An Index Formula for Perturbed Dirac Operators on Lie Manifolds

Abstract

We prove an index formula for a class of Dirac operators coupled with unbounded potentials, also called “Callias-type operators”. More precisely, we study operators of the form $P := \hspace* {.5mm} / \hspace* {-2.3mm}D+ V$, where $\hspace* {.5mm} / \hspace* {-2.3mm}D$ is a Dirac operator and V is an unbounded potential at infinity on a non-compact manifold M ₀. We assume that M ₀ is a Lie manifold with compactification denoted by M. Examples of Lie manifolds are provided by asymptotically Euclidean or asymptotically hyperbolic spaces and many others. The potential V is required to be such that V is invertible outside a compact set K and V ⁻¹ extends to a smooth vector bundle endomorphism over M∖K that vanishes on all faces of M in a controlled way. Using tools from analysis on non-compact Riemannian manifolds, we show that the computation of the index of P reduces to the computation of the index of an elliptic pseudodifferential operator of order zero on M ₀ that is a multiplication operator at infinity. The index formula for P can then be obtained from the results of Carvalho (in K-theory 36(1–2):1–31, 2005). As a first step in the proof, we obtain a similar index formula for general pseudodifferential operators coupled with bounded potentials that are invertible at infinity on a restricted class of Lie manifolds, so-called asymptotically commutative, which includes, for instance, the scattering and double-edge calculi. Our results extend many earlier, particular results on Callias-type operators.

1 Introduction

Perturbed Dirac operators $\hspace* {.5mm}/\hspace* {-2.35mm}D+ V$ and operators Δ+V of Schrödinger type on non-compact manifolds play an important role in quantum mechanics, conformal field theory, and in other areas. Partly for this reason, the index theory for these kinds of operators has been the subject of extensive research [5, 6, 13–15, 18, 29–31, 40, 59].

The purpose of this paper is to give an index formula for $\hspace* {.5mm}/\hspace* {-2.35mm}D+ V$, that is, for a Dirac operator $\hspace* {.5mm}/\hspace* {-2.35mm}D$ coupled with unbounded potential V on an even-dimensional Lie manifold M ₀, a class of non-compact manifolds whose structure at infinity is controlled by a Lie algebra of vector fields tangent to the boundary of a suitable given compactification M of M ₀ (see Sect. 3 and below). We also find an index formula for operators $\hspace* {.5mm}/\hspace* {-2.35mm}D+ V_{0}$ coupled with bounded potentials V ₀ on a subclass of Lie manifolds that are “commutative at infinity” (as in Definition 3.6). One of the points of the calculation is, as in [13], to express the index as the product of two K-classes, one defined by the Dirac operator and the other defined by the potential V using the difference construction as in [7, 39]. See [12, 57, 58] for some recent results on similar problems. See also [33, 46, 63] for applications of these types of index formulas.

Lie manifolds, or manifolds with a Lie structure at infinity, were introduced and studied in [2, 4], extending work by Melrose, Schrohe, Schulze, Vasy, and their collaborators, which in turn build on earlier results by Cordes, Parenti, and others [23, 25, 47, 56, 61]. Many known classes of non-compact manifolds can be given a Lie structure, such as manifolds with cylindrical ends, with conical singularities, asymptotically Euclidean space, hyperbolic space, and manifolds with fibered boundary. There is a natural algebra of differential operators associated with any such manifold that contains all the classical geometric operators, such as the Dirac operator [2]. There is also a well-behaved algebra of pseudodifferential operators on any Lie manifold, which is related to an algebra of pseudodifferential operators on a differentiable groupoid. For manifolds with cylindrical ends, for instance, we have Melrose’s b-calculus [45] and the calculus of Fuchs differential operators. For many of these algebras Ψ ^∗ of pseudodifferential operators on manifolds with corners, the Fredholmness of P∈M _n (Ψ ^∗) can be characterized by the invertibility of a symbol class that consists of two components: the principal symbol σ ₀(P) and a symbol at the boundary σ _∂ (P), also called the indicial operator associated with P. Thus a pseudodifferential operator compatible with the Lie manifold structure is Fredholm if, and only if, the following two conditions are satisfied: the usual ellipticity and the invertibility in the so-called indicial algebra at the boundary. The Fredholm conditions relevant for our case are discussed in Propositions 3.4 and 3.8.

Let M ₀ be an even-dimensional Riemannian Lie manifold, with compactification to a manifold with corners M, and $\mathcal{V}$ be the Lie algebra of vector fields tangent to the faces of M and defining the structure at infinity of M ₀ (for precise definitions, see Sect. 3.1). Let W be a Clifford module over M endowed with an admissible connection and let $\hspace* {.5mm}/\hspace* {-2.35mm}D: \mathcal{C}^{\infty}(M;W) \to \mathcal{C}^{\infty}(M;W)$ be the associated generalized Dirac operator. Let us denote by {x _k } the boundary-defining functions of the hyperfaces of M. Also, let E be a Hermitian bundle endowed with a suitable, compatible (metric-preserving) connection. We shall consider operators of the form

$$ T' = \hspace* {.5mm}/\hspace* {-2.35mm}D+ V := {\hspace* {.5mm}/\hspace* {-2.35mm}D} \widehat {\otimes} 1 + 1 \widehat {\otimes} V: \mathcal{C}^\infty_c(M_0;W\otimes E) \to \mathcal{C}_c^\infty(M_0;W\otimes E), $$

(1)

where the potential $V\in \operatorname {End}(E)$ is of the form V=f ⁻¹ V ₀ with

$$ f := \prod x_k^{a_k}, \quad a_k \in \mathbb {Z},\ a_k > 0, $$

(2)

and V ₀ smooth on M and invertible at infinity (that is, on ∂M). We prove that T′ is essentially self-adjoint acting on L ²(M ₀;W⊗E). We shall denote by T the closure of T′, which is hence a self-adjoint operator (odd with respect to the natural spinor grading). Let $\mathcal{D}(T)$ denote the domain of T and $\mathcal{D}(T) = \mathcal{D}(T)_{+} \oplus \mathcal{D}(T)_{-}$ be its grading. We shall still write $T = \hspace* {.5mm}/\hspace* {-2.35mm}D+ V$, for simplicity. Let $W \widehat{\otimes}E$ be the tensor product W⊗E endowed with the usual grading.

Our main result, Theorem 4.13, is an index formula for the chiral operator

$$ T_+ : \mathcal{D}(T)_+ \to L^2(M_0; W \widehat{\otimes}E)_- $$

similar to the usual Atiyah–Singer index formula. The proof of this theorem is obtained from a sequence of reductions, ultimately reducing our main result to the Atiyah–Singer type theorem for operators that are asymptotically multiplication at infinity [21]. Let us mention that our Theorem 4.13 is about as general as one can hope for such that a classical index formula would still apply. For instance, if one replaces V with a bounded potential V ₀, then one expects an index formula for $\hspace* {.5mm}/\hspace* {-2.35mm}D+ V_{0}$ to involve non-local invariants similar to the eta invariant [8]. These non-local invariants would be associated with the faces at infinity. Thus, if one wants to avoid non-local invariants and have an index formula on an arbitrary Lie manifold just in terms of classical Chern characters, then one needs to require V to be unbounded at infinity. (Note, however, that on asymptotically commutative Lie manifolds, Definition 3.6, we do allow bounded potentials, and the calculation in this case is an important ingredient in the proof; see below for more details.) Moreover, imposing some structure at infinity also seems to be necessary and is usually done in practice. This justifies why we consider Lie manifolds and not more general non-compact manifolds. See [15] and [40] for some related approaches.

Most of the known results on the index of perturbed Dirac operators on non-compact manifolds cited above make use of crucial properties of Dirac operators, namely relative index theorems, trace formulas, or boundary conditions. In this paper, the index formula for $\hspace* {.5mm}/\hspace* {-2.35mm}D+ V$, with V bounded, is obtained from a general index theorem for a suitable class of pseudodifferential operators, and hence holds for perturbed pseudodifferential operators. We need to assume, however, that our Lie manifold is asymptotically commutative (or commutative at infinity), Definition 3.6. A similar approach, in the bounded potentials case, is contained in [40], for odd-dimensional manifolds, where Melrose’s index formula for (families of) scattering operators is used to derive an index theorem for perturbed pseudodifferential operators, so-called Callias-type operators, with bounded potentials. It is shown there that the index can be computed from invariants at the corner $S_{\partial M}^{*}M$. (Note that the scattering structure is just a particular case of the asymptotically commutative Lie structures we consider here. It is easily seen that Theorem 2.5 extends to the case of families, so our results can also be formulated in this setting.) To get the result for unbounded potentials, we need harder results from analysis, so we stick to differential operators, but this result holds for arbitrary Lie manifolds.

Let us now review the sequence of reductions that lead to Theorem 4.13. At the same time, we will review the contents of the paper (but in the reverse order of the sections). The first step is to write

$$ T = \hspace* {.5mm}/\hspace* {-2.35mm}D+ V = f^{-1/2} Q f^{-1/2}, \quad\mbox{ with } Q := f^{1/2} \hspace* {.5mm}/\hspace* {-2.35mm}Df^{1/2} + V_0, $$

which we show in Sect. 4 to have the same index as T. (See also [1, 50] and the comments on the decay of the functions in the kernel of T in Sect. 4.) We then consider a new Lie manifold structure $(M, \mathcal{W})$ on M ₀ using

$$ \mathcal{W}= f\mathcal{V}:= \Bigl(\prod x_k^{a_k}\Bigr)\mathcal{V}. $$

(3)

It turns out that $Q \in \operatorname {Diff}_{\mathcal{W}}(M)$, which justifies the introduction of the new Lie manifold structure $(M, \mathcal{W})$. Moreover, Q itself is a Dirac operator coupled with the bounded potential V ₀. What makes the index of such an operator computable is the fact that the structural Lie algebra of vector fields $\mathcal{W}$ defining Q is commutative at infinity, or, to put this in another way, the indicial algebra of $\mathcal{W}$ is commutative. A Lie manifold with this property will be called asymptotically commutative. The analysis on the new Lie manifold $(M,\mathcal{W})$ turns out to be much easier. The index of the operator $Q = \hspace* {.5mm}/\hspace* {-2.35mm}D+ V_{0}$ associated with (general) asymptotically commutative Lie manifolds with V ₀ bounded, but invertible at infinity, is obtained in Theorem 4.7 using results of Sect. 3.

The analytical properties of general, not necessarily even-dimensional, asymptotically commutative Lie manifolds $(M, \mathcal{W})$ and the index of operators on these spaces are studied in Sect. 3. We show that fully elliptic operators in $\varPsi^{*}(M; \mathcal{W})$ can be deformed continuously to operators in $\varPsi^{*}(M; \mathcal{W})$ that are asymptotically multiplication on M ₀. We thus obtain an index theorem for fully elliptic pseudodifferential operators on general asymptotically commutative Lie manifolds (that is, of the form $(M, \mathcal{W})$ with $\mathcal{W}$ commutative at infinity), Theorem 3.9, generalizing known results for the scattering and double-edge operators [42, 48]. Let us also mention that the case of asymptotically commutative Lie manifolds includes the important case of asymptotically Euclidean manifolds. The index formula for fully elliptic operators on asymptotically commutative Lie manifolds follows then from the results of [21]. This reduction is achieved in Sect. 2. We remark that all the results in Sects. 2 and 3 do not assume that our manifolds are even-dimensional.

It is a classical result that on a compact manifold M ₁, a pseudodifferential operator P of order m defines a Fredholm operator H ^s(M ₁)→H ^s−m(M ₁) if, and only if, it is elliptic. In other words, on a compact manifold, ellipticity is equivalent to Fredholmness. By contrast, on non-compact manifolds, ellipticity is typically only a necessary, but not sufficient condition to ensure Fredholmness; stronger conditions on an operator P are required to obtain that P is Fredholm. For example, on an asymptotically commutative Lie manifold, the Fredholm condition is still controlled by the invertibility of a function, which this time is an extension of the principal symbol, and hence is defined on an extension of the cosphere bundle. This phenomenon is studied in Sect. 2, where an index theorem is proved for such operators by reducing to the case of operators that are multiplication at infinity (which was studied in [21]). In particular, we obtain in that section an index theorem for asymptotically multiplication operators.

We shall assume throughout most of this paper that M ₀ is a non-compact Lie manifold with compactification M, although some of our results of the earlier sections may be true for more general non-compact manifolds. For instance, the index theorem of [21] is valid without any assumption on M ₀.

2 Asymptotically Multiplication Operators

In this section, we review some basic concepts and results to be used in what follows, leading to the Atiyah–Singer index theorem in the setting of non-compact manifolds and operators that are multiplication outside a compact set (or asymptotically so). Here, we keep the manifolds quite general, while we consider a class of operators that naturally inherits the properties of the compact manifold case. For simplicity, we assume that M ₀ is endowed with a metric g and that, as a topological space, it is the interior of a compact manifold with corners M such that TM restricts to TM ₀ on M ₀. We let n be the dimension of M ₀, which in this and the following section may be arbitrary, but in the last section will be assumed to be even.

2.1 General Calculus

We consider for now a smooth manifold M ₀ without boundary, not necessarily compact, and a smooth vector bundle E over M ₀ that is trivial outside a compact set in M ₀. We denote by $d\operatorname {vol}_{g}$ the volume form on M ₀ defined by the metric. We also assume that E is endowed with a Hermitian metric, which is the trivial (product) metric close to infinity. Typically, M ₀ will coincide with the interior of a given compact manifold with corners M.

We first make a short review of main results of the theory of pseudodifferential operators on M ₀ that we need in this paper, in the setting of operators that are multiplication by a smooth function outside a compact set. We closely follow the approach of [21] (see also, however, [34, 38, 62] for general references).

Let us recall that a smooth function p:W×ℝⁿ⟶ℂ^N×N defines a symbol in the class S ^m(W×ℝⁿ) of symbols of order m if, and only if, for any compact set K⊂W and multi-indices α, β, there exists $\mathcal{C}_{K,\alpha,\beta} > 0$ such that $|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}p(x,\xi)|\leq \mathcal{C}_{K,\alpha,\beta}(1+|\xi|)^{m-|\beta|}$, for all x∈K and ξ∈ℝⁿ. An operator $P: \mathcal{C}^{\infty}_{c}(M_{0};E) \to \mathcal{C}^{\infty}(M_{0};E)$ is said to be in the class Ψ ^m(M ₀;E) of pseudodifferential operators of order m on M ₀ if, for any (not necessarily connected) coordinate chart W of M ₀ trivializing E and for any $h_{1}, h_{2}\in \mathcal{C}^{\infty}_{c}(W)$, $h_{1}Ph_{2}: \mathcal{C}_{c}^{\infty}(W)^{N}\to \mathcal{C}_{c}^{\infty}(W)^{N}$ is a matrix of pseudodifferential operators of order m on W, that is, h ₁ P(h ₂ u)=p(x,D)u, with

$$ h_1 P(h_2u)(x) = p(x,D)u(x) := (2\pi)^{-n} \int_{\mathbb {R}^n}{p(x,\xi)\widehat {u}(\xi)e^{ix\cdot\xi} d\xi}, $$

(4)

where $\widehat {u}$ denotes the Fourier transform of u and p∈S ^m(W×ℝⁿ).

We shall work only with classical symbols, that is, symbols that have asymptotic expansions p∼∑p _m−k with p _m−k ∈S ^m−k(W×ℝⁿ) positively homogeneous of degree m−k in ξ. Let us denote by π:T ^∗ M ₀→M ₀ the cotangent bundle of M ₀. The leading term p _m in the expansion of p(x,ξ) as a classical symbol defines a smooth section of the bundle $\operatorname {End}(E)$ over the cotangent bundle T ^∗ M ₀, the principal symbol of P, which is a smooth bundle homomorphism σ _m (P):π ^∗ E→π ^∗ E, positively homogeneous on the fibers of T ^∗ M ₀. By choosing a metric on TM ₀, the class of principal symbols can be identified with $\mathcal{C}^{\infty}(S^{*}M_{0};\operatorname {End}(E))$ where S ^∗ M ₀ is the unit sphere bundle of the cotangent bundle. An operator P is said to be elliptic if σ _m (P) is invertible on S ^∗ M ₀. We shall regard the cosphere bundle S ^∗ M ₀ as the boundary of T ^∗ M ₀ using a radial compactification of each fiber. It is in this sense (using the radial compactification) that we shall often extend the principal symbol of an order zero pseudodifferential operator to T ^∗ M ₀.

Under certain assumptions that will be satisfied in our setting, we have that if P∈Ψ ^m(M ₀;E), then P ₀:=P(1+P ^∗ P)^−1/2∈Ψ ⁰(M ₀;E), with P ^∗ the formal adjoint, and P is Fredholm, respectively, elliptic, if, and only if, P ₀ is, with $\operatorname {ind}(P)=\operatorname {ind}(P_{0})$. Moreover, σ _m (P) is homotopic to σ ₀(P ₀), as sections of S ^∗ M ₀ (the unit sphere bundle of T ^∗ M ₀). Hence, for the purposes of an index formula, we will mainly be concerned with operators of order 0. (Here and below, the Fredholm condition is on the space L ² for the natural metric on the underlying space. As is well known, this definition of the Fredholm property for unbounded operators is equivalent to that of Fredholmness between the suitable Sobolev spaces.)

We start by considering the class of pseudodifferential operators that are multiplication outside a compact subset of M ₀, defined as

(5)

(For m<0, we consider $\varPsi^{{m}}_{\mathrm{mult}}(M_{0};E) := \varPsi^{{0}}_{\mathrm{mult}}(M_{0};E) \cap\varPsi^{{m}}(M_{0};E)$). We have that any operator in $\varPsi^{{0}}_{\mathrm{mult}}(M_{0};E)$ is properly supported and that $\varPsi^{0}_{\mathrm{mult}}(M_{0};E)$ is a ∗-algebra (see [21] for details). Moreover, denoting by $S_{\mathrm{mult}}^{0}(T^{*}M_{0};E)$ the set of bounded symbols in S ⁰(T ^∗ M ₀;E) that are constant on the fibers of T ^∗ M ₀→M ₀ outside a compact subset of M ₀, we have that there is a well-defined symbol map

$$ \sigma_{0}: \varPsi^0_\mathrm{mult}(M_0;E)\to S^0_\mathrm{mult}(T^*M_0;E)/S^{-1}_\mathrm{mult}(T^*M_0;E), $$

(6)

which is a surjective ∗-homomorphism with $\ker(\sigma_{0}) = \varPsi^{-1}_{\mathrm{mult}}(M_{0};E)$, as is the case if M ₀ is compact. Moreover, we have that $P\in\varPsi^{0}_{\mathrm{mult}}(M_{0};E)$ is always bounded as an operator on L ²(M ₀;E), and that $\varPsi^{-1}_{\mathrm{mult}}(M_{0};E)$ consists of compact operators, again as in the classical case of compact manifolds.

We endow the class of symbols $S^{0}_{\mathrm{mult}}(T^{*}M_{0};E)$ with the sup-norm, as a section of $\operatorname {End}(E)$ over T ^∗ M ₀. Note that $S^{0}_{\mathrm{mult}}(T^{*}M_{0};E)$ can be identified with the class of bounded sections in $\mathcal{C}^{\infty}(S^{*}M_{0};\operatorname {End}(E))$ that are constant on the fibers of S ^∗ M ₀ outside a compact K⊂M ₀, and this is consistent with regarding S ^∗ M ₀ as the boundary of (the radial fiberwise compactification of) T ^∗ M ₀.

The class $\mathcal{C}_{a}(S^{*}M_{0};E)$ of asymptotically multiplication symbols is defined as the class of functions $p = p(x,\xi)\in \mathcal{C}(S^{*}M_{0};\operatorname {End}(E))$ such that p(x,ξ) is bounded in the sup-norm and, for all ε>0, there is a compact K _ε ⊂M ₀ such that, for all x∉K _ε ,

$$ \sup\limits_{\xi_1,\xi_2\in S^*_xM_0} \|p(x,\xi_1)-p(x,\xi_2)\|_{\operatorname {End}(E_x)}<\varepsilon. $$

(7)

Roughly speaking, the elements of $\mathcal{C}_{a}(S^{*}M_{0};E)$ are continuous sections of $\operatorname {End}(E)$ over S ^∗ M ₀ that are bounded and asymptotically independent of ξ on the fibers of S ^∗ M ₀. It is easily checked that it is a C ^∗-subalgebra of $\mathcal{C}_{b}(S^{*}M_{0};\operatorname {End}(E))$, the class of continuous, bounded sections of $\operatorname {End}(E)$.

We now define the class of asymptotically multiplication pseudodifferential operators as

$$ \varPsi ^0_{a}(M_0;E) := \overline{ \varPsi ^0_\mathrm {mult}(M_0;E)}\subset \mathcal{B}(L^2(M_0;E)), $$

(8)

that is, the closure of $\varPsi ^{0}_{a}(M_{0};E)$ in the topology of bounded operators on L ²(M ₀;E).

The point of the following lemma is that, once we consider completions, we will need to replace operators that are multiplication at infinity with asymptotically multiplication operators.

Lemma 2.1

The principal symbol defines a natural map $\varPsi^{0}_{\mathrm{mult}}(M_{0};E) \to \mathcal{C}_{a}(S^{*}M_{0}; E)$, which extends by continuity to a surjective map $\varPsi ^{0}_{a}(M_{0};E) \to \mathcal{C}_{a}(S^{*}M_{0};E)$.

Proof

We show that $\mathcal{C}_{a}(S^{*}M_{0};E)$ coincides with the closure of

(9)

and the result then follows as in the compact case. It is readily checked that any p in the closure of (9) is asymptotically multiplication. For the converse, let $p\in \mathcal{C}_{a}(S^{*}M_{0};\operatorname {End}(E))$ and take $\widetilde {p}\in \mathcal{C}_{b}^{\infty}(S^{*}M_{0};\operatorname {End}(E))$ such that $\|\widetilde {p}-p\|_{\sup}<\varepsilon$. Let K⊂M ₀ be compact such that $\|p(x,\xi_{1})-p(x,\xi_{2})\|_{\operatorname {End}(E_{x})}<\varepsilon$, for all x∉K, ξ ₁,ξ ₂∈S ^∗ M ₀, and let $\phi\in \mathcal{C}^{\infty}_{c}(S^{*}M_{0})$ be such that $\operatorname {supp}\phi\subset M_{0}-K$, 0≤ϕ≤1, and ϕ(x,ξ)=1, for x∉K′, with K′ compact such that $K\subset \operatorname{int}(K^{\prime})$. Define

$$q(x,\xi):=(1-\phi(x,\xi))\;\widetilde {p}(x,\xi) + \phi(x,\xi)\;\widetilde {p}(s(x)), $$

where s is a fixed smooth section of S ^∗ M ₀ (which exists since every connected non-compact manifold has a nowhere vanishing vector field). Then $q(x,\xi)\in \mathcal{C}_{b}^{\infty}(S^{*}M_{0};\operatorname {End}(E))$ and for x∉K′, $q(x,\xi)=\widetilde {p}(s(x))$ is independent of ξ. Moreover,

Hence, p lies in the closure of Ξ defined in Eq. (9), and that concludes our proof. □

The following result can be proved much as in the compact case.

Proposition 2.2

The principal symbol map (6) is continuous and the following sequence of C ^∗-algebras is exact:

(10)

where $\sigma_{0}: \varPsi ^{0}_{a}(M_{0};E) \to \mathcal{C}_{a}(S^{*}M_{0};E)$ now denotes the extension by continuity of the classical principal symbol map $\sigma_{0} : \varPsi^{0}_{\mathrm{mult}}(M_{0};E) \to \mathcal{C}_{a}(S^{*}M_{0};E)$.

Proof

The exactness at $\mathcal{C}_{a}(S^{*}M_{0};E)$ follows from Lemma 2.1. Using a partition of unity and the fact that our result is true in the compact case, we see that $\varPsi^{-1}_{\mathrm{mult}}(M_{0};E) \subset \mathcal{K}$ as a dense subset. This proves the exactness at $\mathcal{K}(M_{0}; E)$ and the fact that $\mathcal{K}(M_{0}; E)$ is contained in the kernel of σ ₀.

As in the classical case of compact manifolds, the difficult case is to prove that if an operator $T \in \varPsi ^{0}_{a}(M_{0};E)$ is in the kernel of σ ₀, then it is compact. Let then

$$ T_m \in \varPsi ^0_\mathrm {mult}(M_0;E), \quad T_m \to T \ \ \text{ and } \ \ \sigma_{0}(T_m) \to0. $$

Then we can replace the sequence T _m with a sequence of operators that are zero in a neighborhood of infinity. Also, let $\psi\in \mathcal {C}^{\infty}_{c}(M_{0})$ have the support in a local coordinate chart. Then ψT _m ψ→ψTψ and σ ₀(ψT _m ψ)→0. Using the case of a compact manifold, we see that ψTψ is a compact operator. From this we infer that ψ ₁ Tψ ₂ is also compact for any compactly supported functions ψ ₁ and ψ ₂. (One way to prove this is to first consider the case when ψ ₁ and ψ ₂ have disjoint supports). Let 0≤⋯≤ψ _k ≤ψ _k+1≤⋯≤1 be an increasing sequence of compactly supported functions such that ψ _k (x)→1, for all x. (We are assuming here that M ₀ is σ-compact, which is always the case if M ₀ has a compactification.)

We claim that ψ _k Tψ _k →T. Since ψ _k Tψ _k is compact for any k, it will follow that T is also compact. To prove our claim, let ε>0 and choose m such that ∥T−T _m ∥<ε/3. Then we can find k ₀ such that ∥ψ _k T _m ψ _k −T _n ∥≤ε/3 for k≥k ₀ since T _m is assumed to be zero outside a compact set. Then

$$ \|T - \psi_k T \psi_k\| \le\|T - T_m\| + \| T_m - \psi_k T_m \psi_k\| + \| \psi_k (T_m - T) \psi_k\| \le\varepsilon $$

for k≥k ₀. This completes our proof. □

It follows from Proposition 2.2 that $P\in \varPsi ^{0}_{a}(M_{0};E)$ is a Fredholm operator on L ²(M ₀) if, and only if, its full symbol is invertible in $\mathcal{C}_{a}(S^{*}M_{0};E)$ or, equivalently, in $\mathcal{C}_{b}(S^{*}M_{0};E)$. See also [32] for a discussion of Fredholm operators on non-compact manifolds.

2.2 The Atiyah–Singer Index Theorem

We now review the Atiyah–Singer index formula, applied to asymptotically multiplication operators. (See, for instance, [9, 44] for the details on the constructions below).

First, we define operators acting between sections of two different vector bundles. Let E,F be vector bundles over M ₀, and assume that we are given a fixed isomorphism E≅F outside a compact subset of M ₀ (this assumption is satisfied whenever there exists an elliptic operator acting between these bundles). We then define $\varPsi_{\mathrm{mult}}^{0}(M_{0};E,F)$ as the subclass of $\varPsi _{\mathrm{mult}}^{0}(M_{0};E\oplus F)$ given by those operators that induce $P: \mathcal{C}^{\infty}_{c}(M_{0};E)\to \mathcal{C}^{\infty}(M_{0};F)$. We have also $S^{0}_{\mathrm{mult}}(T^{*}M_{0};E,F)\subset S^{0}_{\mathrm{mult}}(T^{*}M_{0};E\oplus F)$ and all the results in the previous section hold, except that if $P\in \varPsi_{\mathrm{mult}}^{0}(M_{0};E,F)$ then $P^{*}\in\varPsi_{\mathrm{mult}}^{0}(M_{0};F,E)$, so we leave the setting of C ^∗-algebras. In any case, an analogue of the exact sequence given in Proposition 2.2 holds.

We now associate a K-theory class with an elliptic Fredholm operator in $\varPsi_{a}^{0}(M_{0};E,F)$.

Lemma 2.3

For any elliptic, bounded $Q\in\varPsi^{0}_{\mathrm{mult}}(M_{0};E,F)$ such that q:=σ ₀(Q), there is a natural class [σ ₀(Q)]:=[π ^∗ E,π ^∗ F,q] in the compactly supported K-theory of T ^∗ M ₀ obtained by extending q to an invertible map outside a compact set of TM ₀ that is constant along the fibers of TM ₀→M ₀ outside a compact set. This K-theory class is such that the Fredholm index of Q depends only on [σ ₀(Q)].

Proof

In order to associate a K-theory class with a Fredholm operator in $\varPsi_{a}^{0}(M_{0}; E,F)$, we start by noting that if $Q \in\varPsi^{0}_{\mathrm{mult}}(M_{0};E,F)$ is such that q:=σ ₀(Q) is invertible, then q defines an isomorphism outside a compact subset of T ^∗ M ₀ by homogeneity and the fact that it is constant on the fibers outside a compact subset in M ₀. Hence, regarding q(x,ξ) as a bundle map π ^∗ E→π ^∗ F, π:T ^∗ M ₀→M ₀, we obtain the desired definition of [σ ₀(Q)]:=[π ^∗ E,π ^∗ F,q] as in [7, 39]. The dependence of the index only on [σ ₀(Q)] follows as in the classical case by noticing that Q is Fredholm as long as the principal symbol is invertible as in [21]. □

Given now a Fredholm operator $P\in \varPsi ^{0}_{a}(M_{0};E,F)$ with invertible symbol $\sigma_{0}(P)\in \mathcal{C}_{a}(S^{*}M_{0}; E, F)$, by Proposition 2.2, we can take $q\in S^{0}_{\mathrm{mult}}(T^{*}M_{0};E,F)$ sufficiently close to σ ₀(P) such that tσ ₀(P)+(1−t)q, t∈[0,1], is a homotopy through invertible symbols. We define the symbol class of P as

$$ [\sigma_{0}(P)] := [\pi^*E,\pi^*F, q]\in K^0(T^*M_0). $$

(11)

This class is independent of q. If we take q=σ ₀(Q), with $Q \in\varPsi^{0}_{\mathrm{mult}}(M_{0};E,F)$, we have $\operatorname {ind}(P) = \operatorname {ind}(Q)$. Moreover, if two Fredholm operators have the same symbol class, then their indices coincide, and in fact there is a well-defined (analytic) index map

$$ \operatorname {ind}: K^0(TM_0) \to \mathbb {Z}, \quad [\sigma_{0}(P)] \mapsto \operatorname {ind}(P), $$

(12)

where $P\in \varPsi ^{0}_{a}(M_{0};E,F)$ and we use the metric to canonically identify T ^∗ M ₀ with TM ₀. We summarize the above discussion in the following lemma extending Lemma 2.3.

Lemma 2.4

For any elliptic, bounded $Q\in \varPsi ^{0}_{a}(M_{0};E,F)$ such that q:=σ ₀(Q), there is a natural class [σ ₀(Q)]:=[π ^∗ E,π ^∗ F,q] in the compactly supported K-theory of T ^∗ M ₀ obtained by extending q to an invertible map outside a compact set of TM ₀ that is asymptotically constant along the fibers of TM ₀→M ₀. This K-theory class is such that the Fredholm index of Q depends only on [σ ₀(Q)].

For a manifold X, we let H ^∗(X), respectively $H_{c}^{*}(X)$, denote the cohomology, respectively the compactly supported cohomology, with rational coefficients, of X. Recall that throughout this section, we assume that M ₀ is the interior of a compact manifold with corners M. Let $\overline{TM}$ be the radial compactification of the tangent bundle to M. Then the pair $(\overline{TM} , \partial \overline{TM})$ is homeomorphic to the similar pair associated with a manifold with boundary. Hence the (even) Chern character yields a map

$$ \operatorname {ch}_0: K^0(TM_0)\to H_c^{2*}(TM_0) = H^{2*}(\overline{TM}, \partial \overline{TM}). $$

(Later on, we will also consider the odd Chern character $\operatorname {ch}_{1}$ defined on K ¹.) Also, let Td(T _ℂ M)∈H ^∗(M) denote the Todd class of the complexified tangent bundle TM⊗ℂ. Note that since TM is oriented, there is a well-defined fundamental class $[TM_{0}]\in H_{2n}(\overline{TM}, \partial \overline{TM})$ (see, for instance, [44] for details on these constructions).

The following result is an immediate extension of a result in [21] from operators that are multiplication outside a compact subset of M ₀ to operators that are only asymptotically so. Let $\pi: \overline{TM} \to M$ denote the natural projection. We have $\operatorname {ch}_{0}[\sigma_{0}(P)] \in H^{2*}(\overline{TM}, \partial \overline{TM})$ and $\pi^{*}Td{(T_{\mathbb {C}} M)} \in H^{2*}(\overline{TM})$, so their product is in $H^{2*}(\overline{TM}, \partial \overline{TM})=H_{c}^{2*}(TM_{0})$.

Theorem 2.5

Let $P \in \varPsi ^{0}_{a}(M_{0};E,F)$ be such that σ ₀(P) is invertible in $\mathcal{C}_{a}(S^{*}M_{0}; E, F)$. Then P is Fredholm and

$$ \operatorname {ind}(P) = (-1)^n\operatorname {ch}_0[\sigma_{0}(P)]\pi^* Td(T_{\mathbb {C}} M)[TM_0], $$

where [σ ₀(P)] is defined using Lemma 2.4.

Proof

The fact that P is Fredholm follows from Proposition 2.2. The rest of the proof follows from the discussion before the statement of this theorem. Indeed, let $P \in \varPsi ^{0}_{a}(M_{0};E,F)$ be elliptic. Then we can find $P_{0} \in\varPsi^{0}_{\mathrm{mult}}(M_{0};E,F)$ that is close enough to P such that the straight line joining P and P ₀ consists of invertible operators. Both the left-hand side and the right-hand side of our index formula are homotopy invariant. For P ₀ they are equal by [21]. For P, they will therefore be equal as well, by homotopy invariance. □

2.3 Comparison Spaces

In this subsection, we extend the index formula of Theorem 2.5 by deformation to certain pseudodifferential operators on non-compact manifolds that extend to the compactification M of M ₀ in a suitable sense. More precisely, we require the principal symbols of our operators to extend to a so-called “comparison space”, and there is an invertible complete symbol at the boundary. We thus generalize the approach in [31, 48], using homotopy to asymptotically multiplication symbols.

Recall that M is a compactification of M ₀ to a manifold with corners. In this section, we fix a vector bundle A over M such that $A \vert_{M_{0}}\cong TM_{0}$ (later, when we consider Lie structures, such an A will be naturally associated with M ₀.) Denote by $\overline{A}$ the fiber-wise radial compactification of A, so $\overline{A}$ is a manifold with corners that is a fibration over M with fibers closed balls of dimension n. We identify A with A ^∗ using a fixed metric. Let (S ^∗ A) _∂M be the restriction of the cosphere bundle S ^∗ A to the boundary ∂M of M. Define

$$ \varOmega:= \partial(\overline{A}) = (S^*A) \cup\overline{A} \vert_{\partial M} $$

(13)

such that

$$ \mathcal{C}(\varOmega) = \{(f,g) \in \mathcal{C}(S^*{A})\oplus \mathcal{C}( \overline{A} \vert_{\partial M}) : f\vert_{(S^*{A})_{\partial M}} = g_{(S^*{A}) \vert_{\partial M}}\}. $$

The space Ω will play an important role in what follows. It is closely related to a similar space introduced by Cordes and his collaborators in his work on Gelfand theory for non-compact manifolds [23, 24].

Let Ψ _A (M ₀;E)⊂Ψ ⁰(M ₀;E) be a ∗-algebra of order 0, bounded, pseudodifferential operators. We say that Ω is a comparison space for Ψ _A (M ₀;E) if there is a surjective homomorphism

$$ \sigma_\mathrm{full} : \overline{\varPsi_A(M_0;E)} \to \mathcal{C}(\varOmega; { \operatorname {End}(E) } ) $$

(14)

such that $\sigma_{\mathrm{full}}(P)\vert_{S^{*}M_{0}} = \sigma_{0}(P)$, with kernel included in the algebra of compact operators. We call σ _full a full symbol and write σ _full=(σ ₀,σ _∂ ), where

$$ \sigma_\partial : \overline{ \varPsi_A(M_0;E) } \to \mathcal{C}(\overline{A} \vert_{\partial M}; \operatorname {End}(E) \vert_{\partial M}) $$

is the boundary symbol morphism. An operator with invertible full symbol is called fully elliptic. We shall give an index formula for fully elliptic operators in this setting, reducing to asymptotically multiplication operators.

We see first that any function in $\mathcal{C}(\varOmega, \operatorname {End}(E))$ can be homotoped over the interior to an asymptotically multiplication symbol. Since the fibers of $\overline{A}$ are isomorphic to the n-dimensional half sphere $\mathbb{S}^{n}_{+}$ and hence contractible, we have that Ω is homotopy equivalent to the space $\widetilde {\varOmega}$ obtained from the cosphere bundle S ^∗ A by collapsing the fibers above points of the boundary. More precisely, $\widetilde {\varOmega}:=(S^{*}{A}) \slash\sim$, with (x,ξ)∼(x,ξ′), for x∈∂M, ξ,ξ′∈S ^∗ A _x . Since

we conclude that every $f\in \mathcal{C}(\varOmega;\operatorname {End}(E))$ is canonically homotopic to some $\widetilde {f}\in \mathcal{C}(\varOmega;\operatorname {End}(E))$ constant on the fibers of $\overline{A}\vert_{\partial M} \to\partial M$ (this is achieved by a homotopy equivalence between Ω and $\widetilde{\varOmega}$). Moreover, if f is invertible, the canonical homotopy between f and $\widetilde {f}$ is through invertible functions. We now have the following:

Lemma 2.6

If $f\in \mathcal{C}(\varOmega, \operatorname {End}(E))$ is constant on fibers of $\overline{A}\vert_{\partial M} \to\partial M$ and $f_{0} := f\vert_{ S^{*}M_{0}}$, then $f_{0} \in \mathcal{C}_{a}(S^{*}M_{0}, \operatorname {End}(E))$. In particular, let $f\in \mathcal{C}(\varOmega, \operatorname {End}(E))$, then f is homotopic to $\widetilde {f} \in \mathcal{C}(\widetilde {\varOmega}, \operatorname {End}(E)) \subset \mathcal{C}(\varOmega , \operatorname {End}(E))$ and hence it satisfies $f_{0} := \widetilde {f}\vert_{ S^{*}M_{0}} \in \mathcal{C}_{a}(S^{*}M_{0}, \operatorname {End}(E))$.

Proof

We consider only the scalar case (so $E=\mathbb {C}= \operatorname {End}(E)$) for simplicity. Let $p\in \mathcal{C}(\partial M)$ be such that $f_{(S^{*}{A})_{\partial M}}=p$. It suffices to show that, given ε>0, there is a neighborhood U of ∂M in M such that for y∈M ₀∩U, $\xi, \xi^{\prime}\in S^{*}_{y}M_{0}$,

$$ \|f_0(y,\xi)-f_0(y, \xi^\prime)\|<\varepsilon $$

(15)

so that (7) follows with K=M∖U. We can do this locally, so assume U _x is a neighborhood of x∈∂M such that π ⁻¹(U _x )≅U _x ×S ⁿ⁻¹, π:S ^∗ A→M. Let

$$ V_x := \{(y,\xi)\in U_x\times S^{n-1} \colon\|f(y,\xi)-p(x)\|<\varepsilon\slash2\}. $$

By the continuity of f in S ^∗ A, we have that V _x is open in U _x ×S ⁿ⁻¹. Moreover, since f(y,ξ)=p(y), y∈∂M, it contains W _x ×S ⁿ⁻¹ for some W _x ⊂∂M open. Hence, V _x  can be taken as $\widetilde {U}_{x}\times S^{n-1}$, for some open neighborhood $\widetilde {U}_{x}$ of W _x in M so that (15) holds if $y\in \widetilde {U}_{x}$. The last statement follows from the fact that Ω and $\widetilde {\varOmega}$ are homotopy equivalent. □

Let us notice that for $f \in \mathcal{C}(\widetilde{\varOmega}; \operatorname {End}(E))$, the restriction $f\vert_{S^{*}M_{0}}$ completely determines f since S ^∗ M ₀ is dense in $\widetilde{\varOmega}$. Now let $P\in\overline{ \varPsi_{A}^{0}(M;E)}$ have invertible symbol

$$ \phi= (\sigma_{0}(P), \sigma_{\partial }(P)) \in \mathcal{C}(\varOmega;\operatorname {End}(E)). $$

(In the terminology introduced earlier, P is fully elliptic.) From the previous lemma, there is an invertible $\widetilde {\phi} \in \mathcal{C}(\widetilde {\varOmega}; \operatorname {End}(E)) \subset \mathcal{C}(\varOmega; \operatorname {End}(E))$ homotopic through invertibles to ϕ, with $\widetilde {\sigma} := \widetilde {\phi}\vert_{ S^{*}M_{0}} \in \mathcal{C}_{a}(S^{*}M_{0}; \operatorname {End}(E))$ and σ:=σ ₀(P) and $\widetilde {\sigma}$ are homotopic (over M ₀). If we let $\widetilde {\sigma} = \sigma_{0}(\widetilde {P})$, for some $\widetilde {P} \in\varPsi^{0}_{a}(M_{0};E) \cap\overline{ \varPsi_{A}^{0}(M;E)}$, then $\widetilde {P}$ is Fredholm, since $\widetilde {\sigma}$ is invertible. Moreover, from the surjectivity of the complete symbol map (14), there is a continuous family $P_{t}\in\overline{ \varPsi_{A}^{0}(M;E)}$, t∈[0,1], lifting the homotopy between ϕ and $\widetilde{\phi}$. Hence P ₀=P and $P_{1}=\widetilde {P}$ and $\operatorname {ind}(P) = \operatorname {ind}(\widetilde {P})$.

We also conclude that with any fully elliptic operator $P \in \overline{ \varPsi_{A}^{0}(M;E, F)}$ we can associate a well-defined K-theory class $[\widetilde {\sigma}_{\mathrm{full}}(P)]$ extending the definition in Eq. (11) by homotopy as follows. We know that P is homotopic to some $\widetilde{P} \in\varPsi^{0}_{a}(M_{0};E)$ through Fredholm operators in $\varPsi^{0}_{A}(M_{0};E)$ and $[\widetilde {\sigma}_{\mathrm{full}}(P)] := [\sigma_{0}(\widetilde{P})]$, so that

$$ [\widetilde {\sigma}_\mathrm{full}(P)] := [\sigma_0(\widetilde {P})] = [\pi^*E, \pi^*F, \widetilde{p}] \in K^0(TM_0), $$

(16)

where we assume $\widetilde{p}$ is an extension of the principal symbol of $\widetilde{P}$ to a function that is multiplication at infinity and homotopic to σ ₀(P) over the interior. In particular,

$$ \operatorname {ind}(P) = \operatorname {ind}(\widetilde{P}) = \operatorname {ind}([\sigma_0(\widetilde {P})]) = \operatorname {ind}([\widetilde {\sigma}_\mathrm{full}(P)]). $$

Now consider, for $P \in\overline{ \varPsi_{A}^{0}(M;E, F)}$ fully elliptic,

$$ [\sigma_\mathrm{full}(P)] := [(\sigma_0(P), \sigma_\partial (P))]\in K_1(\mathcal{C}(\varOmega))\cong K^1(\varOmega), $$

(17)

where $\sigma_{\partial }(P)\in \mathcal{C}(\overline{A}_{\partial M}; \operatorname {Hom}(E,F) \vert _{\partial M})$ and $\sigma_{0}(P)\in \mathcal{C}(S^{*}A;\operatorname {Hom}(E,F))$ denote the boundary and principal symbols, respectively. Let us consider the connecting map ∂:K ¹(Ω)→K ⁰(TM ₀) in the long exact sequence of the pair $(\overline{A}, \partial \overline{A} )=(\overline{A}, \varOmega)$. We summarize the above discussion to the following generalization of Lemma 2.4.

Lemma 2.7

For any fully elliptic $P \in\overline{ \varPsi_{A}^{0}(M;E, F)}$ there is a natural class

$$ [\widetilde {\sigma}_\mathrm{full}(P)] := [\pi^*E,\pi^*F, p] $$

in the compactly supported K-theory of T ^∗ M ₀ obtained by extending σ _full(P) to a continuous endomorphism p invertible outside a compact set of TM ₀. This K-theory class is such that the Fredholm index of P depends only on $[\widetilde {\sigma}_{\mathrm{full}}(P)]$ and

$$ [\widetilde {\sigma}_\mathrm{full}(P)] = \partial [(\sigma_0(P), \sigma_\partial (P))]=\partial [\sigma_\mathrm{full}(P)]. $$

Let us also notice that Ω is homotopically equivalent to the boundary of an oriented smooth manifold with boundary, obtained from $\overline{A}$ by smoothing the corners (so the homotopy equivalence is not C ^∞). Hence it has a well-defined fundamental class [Ω]∈H _2n−1(Ω). If $[\overline{A}]$ denotes the fundamental class of $\overline{A}$ in $H_{2n}(\overline{A}, \varOmega)$, then $[\varOmega] = \partial [\overline{A}]$. Using the compatibility of the boundary maps in K-theory and cohomology, that is, the fact that the Chern character is a natural transformation of cohomology theories (see [55] for an extension of this result to non-commutative algebras), we obtain the following result as a consequence of the Atiyah–Singer index formula extended to operators that are asymptotically multiplication operators (Theorem 2.5).

As before, let Td(T _ℂ M) denote the Todd class of the complexified tangent bundle of M, and $\pi: \overline{T^{*}M}\to M$. Also, we denote by π _Ω :Ω→M the natural projection.

Theorem 2.8

Let Ω be a comparison space for $\varPsi_{A}^{0}(M;E, F)$ and $P \in\overline{\varPsi_{A}^{0}(M;E, F)}$ be a fully elliptic operator (that is, an elliptic operator with σ _∂ (P) invertible in $\mathcal{C}(\overline{A}\vert_{\partial M};\operatorname {Hom}(E, F) \vert_{\partial M})$). Then P is Fredholm and

where $[\widetilde {\sigma}_{\mathrm{full}}(P)]$ is defined using Lemma 2.7.

Proof

Again, the proof of the first equality follows from the discussion before the statement of the theorem. Indeed, let us choose a homotopy between P and $\widetilde{P} \in\varPsi_{a}^{0}(M_{0}; E, F)$ through Fredholm operators $\overline{\varPsi_{A}^{0}(M;E, F)}$. Both the left-hand side and the right-hand side(s) of the index formula of this theorem are homotopy invariant. For $\widetilde{P}$ they are equal in view of Theorem 2.5, by homotopy invariance, they will be equal also for P. To prove the last equality, we just use the fact that the Chern character is compatible with the boundary maps in K-theory and cohomology. (A proof of a generalization of this result to non-commutative algebras can be found in [55].) □

3 Index Formula on Asymptotically Commutative Lie Manifolds

From now on, we endow M ₀ with the structure of a Lie manifold with compactification M and structural Lie algebra of vector fields $\mathcal{V}$ (see below for the definitions). There is associated with $(M, \mathcal{V})$ a well-behaved pseudodifferential calculus and, for operators in this calculus, Fredholm criteria follow from the pseudodifferential calculus of operators on groupoids [2, 41]. See also [19, 20, 22, 27, 28, 49, 53] for more index theory results involving groupoids.

We show that if we introduce the additional assumption on the structural Lie algebra $\mathcal{V}$ that it be asymptotically commutative, then there will exist a (commutative) complete symbol and hence we can apply the results in the previous section. Recall that in this section n may be arbitrary (in the following section, it will be assumed to be even).

3.1 Operators on Lie Manifolds

In this section, M will denote a compact manifold with corners and $M_{0} = \operatorname{int}(M)$, as before. Also, let ${\mathcal{V}}_{M}$ denote the Lie algebra of vector fields that are tangent to all faces of M. We always assume that each hyperface H⊂M is an embedded submanifold of M and hence that it has a defining function x _H (recall that this means that x _H is smooth on M, x _H ≥0, H={x _H =0}, and dx _H ≠0 on H).

We recall the main definitions of [2, 4]. We say that a Lie subalgebra $\mathcal{V}\subset{ \mathcal{V}_{M}}$ is a structural Lie algebra of vector fields if it is a Lie algebra with respect to the Lie bracket and it is also a finitely generated, projective, $\mathcal{C}^{\infty}(M)$-module. By the Serre–Swan theorem, we have that there exists a vector bundle A such that $\mathcal{V}\cong \varGamma( A)$. Moreover, there is a vector bundle morphism ρ:A→TM, called the anchor map, which induces the inclusion map $\rho: \mathcal{V}= \varGamma(A) \to\varGamma(TM)$. It thus follows that A with the given structure is naturally a Lie algebroid.

Definition 3.1

A Lie manifold M ₀ is given by a pair $(M,\mathcal{V})$, where $M_{0}=\operatorname{int}(M)$ and $\mathcal{V}$ is a structural Lie algebra of vector fields such that $\rho\vert_{M_{0}} : A\vert_{M_{0}}\to TM_{0}$ is an isomorphism.

A metric on M ₀ that is obtained from a metric on A by restriction to $A\vert_{M_{0}} \cong TM_{0}$ will be called a compatible metric on M ₀. Any two such metrics are Lipschitz equivalent. We fix one of these metrics on M ₀ in what follows.

With a Lie manifold $(M, \mathcal{V})$ we associate the algebra $\operatorname {Diff}_{\mathcal{V}}(M_{0})$ of $\mathcal{V}$-differential operators on M ₀, defined as the enveloping algebra of $\mathcal{V}$ (generated by $\mathcal{V}$ and $\mathcal{C}^{\infty}(M)$). It was shown in [4] that $\operatorname {Diff}_{\mathcal{V}}(M_{0})$ contains all geometric operators on M ₀ associated with a compatible metric, such as the Dirac and generalized Dirac operators. (This property of $\operatorname {Diff}_{\mathcal{V}}(M_{0})$ will be used in Sect. 4.) One defines differential operators acting between sections of vector bundles E,F over M as

$$ \operatorname {Diff}_{\mathcal{V}}(M_0;E,F):= e_FM_N(\operatorname {Diff}_{\mathcal{V}}(M_0))e_E, $$

where e _E ,e _F are projections onto E,F⊂M×ℂ^N. In [2], a class of pseudodifferential operators associated with a given Lie structure at infinity is defined by a process of microlocalizing $\operatorname {Diff}_{\mathcal{V}}(M_{0};E,F)$. We outline this construction below.

Recall that we first define the class S ^m(A ^∗)⊂C ^∞(A ^∗) as functions satisfying the usual symbol estimates on coordinate patches trivializing A ^∗, which are moreover classical symbols. By inverse Fourier transform on the fibers, each symbol a∈S ^m(A ^∗) defines a distribution $\mathcal{F}_{\mathrm{fib}}^{-1}(a)$ on A that is conormal to M. By restriction, $\mathcal{F}_{\mathrm{fib}}^{-1}(a)$ defines a distribution on TM ₀ conormal to M ₀. We fix a metric on A which then defines a compatible metric. We denote by exp the (geodesic) exponential map associated with this metric (yielding exp _x :T _x M ₀→M ₀ for each x∈M ₀). Now for some r>0, let

$$ \varPhi: (TM_0)_r \to V_r \subset M_0 \times M_0,\quad v \in (T_x M_0)_r \mapsto(x, \exp_x(-v)) $$

be the diffeomorphism given by the Riemann–Weyl fibration, where (TM ₀) _r are the vectors with norm less than r, V _r is an open neighborhood of the diagonal $M_{0} \cong\varDelta_{M_{0}} \subset M_{0}^{2}$, and r>0 is less than the injectivity radius of M ₀, which is known to be positive. Fix a smooth function χ, with $\operatorname {supp}\chi\subset\varPhi^{-1}(V_{r})$ and χ=1 on a neighborhood of the zero section of A, which is identified with M. For a∈S ^m(A ^∗), define a distribution on $M_{0}^{2}$, conormal to M ₀, by

$$ q_\chi(a) := \varPhi_*( \chi \mathcal{F}^{-1}_\mathrm{fib}(a)). $$

(18)

Let a _χ (D) denote the operator on M ₀ with Schwartz kernel q _χ . Then a _χ (D) is a properly supported (if r<∞) pseudodifferential operator on M ₀.

For each $X\in \mathcal{V}=\varGamma(A)$, let $\psi_{X}: \mathcal{C}^{\infty}_{c}(M)\to \mathcal{C}^{\infty}_{c}(M)$ be the operator induced by the global flow Ψ _X :ℝ×M→M by evaluation at 1.

Definition 3.2

The space $\varPsi_{\mathcal{V}}^{m}(M_{0})$ of pseudodifferential operators generated by the Lie structure at infinity $(M,\mathcal{V})$ is the linear space of operators $\mathcal{C}^{\infty}_{c}(M_{0})\to \mathcal{C}^{\infty}_{c}(M_{0})$ generated by a _χ (D), with a∈S ^m(A ^∗) and $b_{\chi}(D) \psi_{X_{1}}\cdots\psi_{X_{k}}$, with b∈S ^−∞(A ^∗) and $X_{1}, \ldots, X_{k}\in \mathcal{V}=\varGamma(A)$.

We similarly define the space $\varPsi^{m}_{\mathcal{V}}(M_{0};E,F)$ of pseudodifferential operators acting between sections of vector bundles E,F over M.

As for the usual algebras of pseudodifferential operators, we have the following basic property of the principal symbol (Proposition 2.6 [2]): The principal symbol establishes isomorphisms

$$ \sigma_m : \varPsi^m_\mathcal{V}(M_0)/\varPsi^{m-1}_\mathcal{V}(M_0)\to S^m(A^*)/S^{m-1}(A^*)\cong \mathcal{C}^\infty(S^*A). $$

We note that the set of all a _χ (D), with a∈S ^∞(A ^∗) is not closed under composition of operators, which is why we consider extra operators in Ψ ^−∞(M ₀). To show that $\varPsi_{\mathcal{V}}^{m}(M_{0})$ is indeed closed under composition, results from the following section are needed.

3.2 Operators on Groupoids

In order to obtain algebraic properties and, in particular, Fredholm criteria, an important result is that $\varPsi_{\mathcal{V}}^{m}(M_{0})$ can be recovered from an algebra of pseudodifferential operators on a suitable groupoid integrating A. We review the main definitions of the theory of pseudodifferential operators on groupoids, for the benefit of the reader (see [41, 54]).

For a Lie groupoid $\mathcal{G}$ with the space of units given by a manifold with corners M, with $d,r: \mathcal{G}\to M$ the domain and range maps, $P=(P_{x}) \in\varPsi^{m}(\mathcal{G})$ is defined as a smooth family of pseudodifferential operators on the fibers $\mathcal{G}_{x}:=d^{-1}(x)$, x∈M, which is right-invariant, that is, U _g P _d(g)=P _r(g) U _g , where $U_{g}: C^{\infty}(\mathcal{G}_{d(g)}) \to C^{\infty}(\mathcal{G}_{r(g)}), U_{g}(f)g^{\prime}:= f(g^{\prime}g)$. Recall that the definition of a Lie groupoid requires the sets $\mathcal{G}_{x} := d^{-1}(x)$ to be smooth manifolds (no corners). We also assume that this family is uniformly supported in that

$$ \operatorname {supp}(P) = \overline{\bigcup_x\mu(\operatorname {supp}(K_x))}\subset \mathcal{G}$$

is compact, where μ(g,h)=gh ⁻¹ and K _x denotes the Schwartz kernel of P _x (a distribution on $\mathcal{G}_{x}\times \mathcal{G}_{x}$). In this case, each P _x is properly supported, so that the composition gh ⁻¹ is well defined. Moreover, P acts on $C^{\infty}(\mathcal{G})$. Let $T^{d}\mathcal{G}=\ker d^{*}=\bigcup T_{x}\mathcal{G}_{x}$ be the d-vertical tangent bundle and denote by $A(\mathcal{G}):=\left(T^{d}\mathcal{G}\right)_{M}$ the Lie algebroid of $\mathcal{G}$ and $S^{*}A(\mathcal{G}):=(A(\mathcal{G})^{*} \setminus 0) \slash \mathbb {R}^{*}_{+}$ its cosphere bundle. Let us fix a metric on A. This choice defines a principal symbol map $\sigma_{m}: \varPsi^{m}(\mathcal{G}) \to \mathcal {C}^{\infty}_{c}(S^{*}A(\mathcal{G}))$, which is surjective, with kernel $\varPsi^{m-1}(\mathcal{G})$. One can similarly define operators acting between sections of vector bundles: If E is a vector bundle over the space of units M, then $\varPsi^{m}(\mathcal{G}, r^{*}E)$ is well defined, as above.

For each x∈M, we consider the regular representation π _x of $\varPsi^{\infty}(\mathcal{G})$ on ${\mathcal{C}^{\infty}(\mathcal{G}_{x})}$ defined as π _x (P):=P _x . When restricted to order zero operators, this is a bounded ∗-representation, for all x. For $P\in\varPsi^{0}(\mathcal{G})$, let

$$ \|P\|_ r := \sup_{x\in M} \|\pi_x(P)\| $$

(19)

be the reduced C ^∗ -norm. We shall also need the full C ^∗ -norm defined as

$$ \|P\|:=\sup_\rho\|\rho(P)\|, $$

where ρ ranges through bounded ∗-representation of $\varPsi^{0}(\mathcal{G})$ such that for $T\in\varPsi^{-\infty}(\mathcal{G})$, ρ(T)≤∥T∥₁, with ∥ ∥₁ defined by integrating the Schwartz kernels over the fibers (see [41] for the precise definitions). Endowing $\varPsi^{0}(\mathcal{G})$ with the full norm ∥ ∥, we have that the principal symbol extends to a bounded ∗-homomorphism

$$ \sigma_0 : \overline{ \varPsi^0(\mathcal{G})} \to \mathcal{C}_0(S^*A(\mathcal{G})), $$

(20)

surjective, with kernel $C^{*}(\mathcal{G}) : = \overline{\varPsi^{-\infty}(\mathcal{G})}$. (A similar result holds for the reduced norm.)

If Y⊂M is an invariant subset of the units, in the sense that r ⁻¹(Y)=d ⁻¹(Y), then $\mathcal{G}_{Y}:=d^{-1}(Y)$ is also a continuous family groupoid, with units Y, and we can define a restriction operator $\mathcal{R}_{Y}: \varPsi^{m}(\mathcal{G};E)\to\varPsi^{m}(\mathcal{G}_{Y}; E_{Y})$ as

$$ P = (P_x)_{x\in M} \to \mathcal{R}_Y(P) = (P_x)_{x \in Y}. $$

(21)

In this case, Lemma 3 in [41] gives that the following sequence is exact:

(22)

where the fibered product $\overline{\varPsi^{0}(\mathcal{G}_{Y})} \times_{\mathcal{C}_{0}(S^{*}A(\mathcal{G})_{Y})} \mathcal{C}_{0}(S^{*}A(\mathcal{G}))$ is defined as the algebra of pairs $(Q, f) \in\overline{\varPsi^{0}(\mathcal{G}_{Y})} \times \mathcal{C}_{0}(S^{*}A(\mathcal{G}))$ such that $\sigma_{0}(Q)=f_{\vert{S^{*}A(\mathcal{G})_{Y}}}$.

If $M_{0}=\operatorname{int} (M )$ is an invariant subset, one can define the so-called vector representation $\pi_{M_{0}}$, which associates with $P\in\varPsi^{0}(\mathcal{G})$ a pseudodifferential operator $\pi_{M_{0}}(P): \mathcal{C}^{\infty}_{c}(M_{0})\to \mathcal{C}^{\infty}_{c}(M_{0})$ by the formula $\pi_{M_{0}}(P)u = u_{0}$, with P(u∘r)=u ₀∘r [43]. Recall that a Lie groupoid is called d-connected if all the sets $\mathcal{G}_{x} := d^{-1}(x)$ are connected. If A→M is a Lie algebroid on M, we say that $\mathcal{G}$ integrates A if $A(\mathcal{G}) = A$. We shall need Theorem 3.3 from [2], which gives that

Theorem 3.3

Let $(M,\mathcal{V})$ be a Lie manifold with Lie algebroid A and $\mathcal{G}$ be a d-connected groupoid over M integrating A. Then $\varPsi_{\mathcal{V}}^{m}(M_{0})\cong\pi_{M_{0}}(\varPsi^{m}(\mathcal{G}))$.

The right-hand side is well defined since, as we shall see next, one can always assume that M ₀ is an invariant subset of such $\mathcal{G}$. In particular, it follows that the classes $\varPsi_{\mathcal{V}}^{m}(M_{0})$ define a filtered algebra on $\varPsi_{\mathcal{V}}^{\infty}(M_{0})$.

The problem of integrating Lie algebroids was solved in [26], though for our purposes, the results in [52] suffice. Namely, M ₀ and ∂M form an A-invariant stratification of M, so it follows from the glueing theorem in [52] that it suffices to integrate along these strata. Since the anchor map ρ is a diffeomorphism over the interior, we can take the d-connected groupoid $\mathcal{G}$ integrating A to coincide with the pair groupoid over the interior, meaning that $\mathcal{G}_{M_{0}}\cong M_{0}\times M_{0}$, in case M ₀ is connected. (The general case of non-connected M ₀ can be reduced to the connected case by taking the compactification of each connected component.) It then follows from [52] that, if $\mathcal{G}_{\partial M}$ is a groupoid integrating A _∂M , then

$$ \mathcal{G}= \mathcal{G}_{M_0}\sqcup \mathcal{G}_{\partial M} \cong(M_0\times M_0) \cup \mathcal{G}_{\partial M} $$

(23)

has the structure of a differentiable groupoid with Lie algebroid A. We see that M ₀ is indeed an invariant subset, and moreover, since $\mathcal{G}_{M_{0}}$ is the pair groupoid, one has that $C^{*}(\mathcal{G}_{M_{0}})\cong \mathcal{K}(L^{2}(M_{0}))$, the isomorphism being induced either by the vector representation $\pi_{M_{0}}$ or by π _x , x∈M ₀, noting that these representations are equivalent through the isometry $r: \mathcal{G}_{x}\to M_{0}$. In particular, the vector representation $\pi_{M_{0}}$ is bounded. Fredholm criteria now follow from the exact sequence (22) as in [41, Theorem 4].

From now on we shall assume that $\mathcal{G}$ is a d-connected Lie groupoid integrating the Lie algebroid A→M defined by a Lie manifold $(M, \mathcal{V})$ . We shall also assume that the vector representation $\pi_{M_{0}}$ is injective on $\overline{\varPsi^{0}(\mathcal{G})}$ . In particular, $\varPsi^{0}(\mathcal{G}) \cong\varPsi^{0}_{\mathcal{V}}(M_{0})$. Moreover, since $C^{*}(\mathcal{G}) \subset\overline{\varPsi^{0}(\mathcal{G})}$ and $\pi_{M_{0}}$ factors through the reduced C ^∗-algebra of $\mathcal{G}$, so that $\pi_{M_{0}}(C^{*}_{r}(\mathcal{G}))=\pi_{M_{0}}(C^{*}(\mathcal{G}))$, we hence obtain that $\mathcal{G}$ is amenable, in that the reduced and full norms coincide.

We shall use the isomorphism above to carry to $\varPsi^{m}_{\mathcal{V}}(M_{0})$ all concepts defined for $\varPsi^{m}(\mathcal{G})$. At the level of symbols, we have $\sigma_{m}(P)=\sigma_{m}(\pi_{M_{0}}(Q))=\sigma_{m}(Q)$ on M ₀, for any $P \in\varPsi^{m}_{\mathcal{V}}(M_{0})$. We shall also need the map of restriction to the boundary for operators on $(M,\mathcal{V})$

$$ \sigma_\partial : \overline{\varPsi^0_{\mathcal{V}}(M_0)} \to\overline{\varPsi^0 (\mathcal{G}_{\partial M})}, \quad P\mapsto \mathcal{R}_{\partial }(Q) = Q\vert_{\partial M}, $$

where $\pi_{M_{0}}(Q)=P$ and $\mathcal{R}_{\partial }: \varPsi^{0}(\mathcal{G})\to\varPsi^{0}(\mathcal{G}_{\partial M})$ is restriction to the boundary, as in Eq. (21).

Proposition 3.4

Let $(M,\mathcal{V})$ be a Lie manifold with Lie algebroid A and $\mathcal{G}$ be a d-connected groupoid as in (23) satisfying $A(\mathcal{G}) \simeq A$. Assume that the representation $\pi_{M_{0}}$ is injective on $\overline{\varPsi^{0}(\mathcal{G})}$, as above. Then

and $P \in\overline{\varPsi_{\mathcal{V}}^{0}(M_{0})}$ is Fredholm on L ²(M ₀) if, and only if, it is elliptic and σ _∂ (P) is invertible in $\overline{\varPsi^{0}(\mathcal{G}_{\partial M})}$.

Proof

Since $\pi_{M_{0}}$ is injective and $\pi_{M_{0}}C^{*}(\mathcal{G}_{M_{0}})\cong \mathcal{K}(L^{2}(M_{0}))$, we have the induced representation $\pi': \overline{\varPsi^{0}(\mathcal{G})} /C^{*}(\mathcal{G}_{M_{0}})\to {\mathcal{B}(L^{2}(M_{0}))}/\mathcal{K}$, which is also injective. Hence, $P=\pi_{M_{0}}(Q)$ is Fredholm if, and only if, the class of Q is invertible in $\overline{\varPsi^{0}(\mathcal{G})} /C^{*}(\mathcal{G}_{M_{0}})$. The result follows from (22). □

Moreover, the amenability of $\mathcal{G}$ yields that the restriction $\mathcal{G}_{\partial M}$ is also amenable (see [60] Proposition 3.7). In this case, ρ:=∏ _x∈∂M π _x is an injective representation of $\varPsi^{0}(\mathcal{G}_{\partial M})$ and σ _∂ (P), as above, is invertible if, and only if, σ _∂ (P) _x =Q _x is invertible for all x∈∂M, with $\pi_{M_{0}}(Q)=P$. (The same is also true for $\varPsi^{\infty}(\mathcal{G}_{\partial M})$, since if ρ(P)=0, then ρ(P(1+P ^∗ P)^−1/2)=0.)

Elliptic operators P with invertible σ _∂ (P) are sometimes called fully elliptic and the algebra $\varPsi^{0}(\mathcal{G}_{\partial M})$ is the so-called indicial algebra. If $\pi_{M_{0}}$ is not injective for some x∈M ₀, then we only have a sufficient condition for Fredholmness.

To finish this section, we prove a result that will later enable us to compute the index of operators with order m>0 from the index of order 0 operators.

Lemma 3.5

Let $Q \in\varPsi^{m}_{\mathcal{V}}(M_{0};E)$ and P:=Q(1+Q ^∗ Q)^−1/2. Then $P \in\overline{\varPsi^{0}_{\mathcal{V}}(M_{0}; E)}$.

Proof

Let $\mathcal{G}$ be the canonical groupoid integrating $(M, \mathcal{V})$. It follows from groupoid calculus applied to $\varPsi^{0}(\mathcal{G})$, more precisely from Theorem 7.2 in [43], that if $L \in{\varPsi^{2m}(\mathcal{G})}$ is such that L≥1 and σ _2m (L)>0 then $SL^{-1/2} \in \overline{\varPsi^{0}(\mathcal{G})}$, for any $S\in\varPsi^{m}(\mathcal{G})$. From Theorem 3.3, let $R \in{\varPsi^{m}(\mathcal{G})}$ be such that $\pi_{M_{0}}(R) = Q$, with $\pi_{M_{0}}$ the vector representation. Then

$$ 1 + Q^*Q = 1 + \pi_{M_0}(R)^*\pi_{M_0}(R) = \pi_{M_0}(1+R^*R) $$

and we can apply the result above to 1+R ^∗ R to obtain $R(1+R^{*}R)^{-1/2}\in\overline{{ \varPsi}^{0}(\mathcal{G})}$. Hence $\pi_{M_{0}}(R(1+R^{*}R)^{-1/2}) \in\overline{{ \varPsi_{\mathcal{V}}}^{0}(M_{0})}$. It follows from the definitions that

$$ \pi_{M_0}((1+R^*R)^{-1/2}) = \pi_{M_0}(1+R^*R)^{-1/2}, $$

so that $P = Q(1+Q^{*}Q)^{-1/2} \in\overline{\varPsi^{0}_{\mathcal{V}}(M_{0})}$. □

Note that if P and Q are as in the lemma above (Lemma 3.5), σ _m (Q) and σ ₀(P) are homotopic as sections of S ^∗ A. Moreover, if we define the map of restriction to the boundary $\sigma_{\partial }: \varPsi^{m}_{\mathcal{V}}(M_{0};E)\to\varPsi^{m}(\mathcal{G}_{\partial M}; r^{*}E)$ given, as before, by $\sigma_{\partial }(Q):=\mathcal{R}_{\partial }(R)$, with $\pi_{M_{0}}(R)=Q$, then it follows from the proof that

$$ \sigma_\partial (P) = \sigma_\partial (Q(1+Q^*Q)^{-1/2}) = \sigma_\partial (Q) (1 + \sigma_\partial (Q)^*\sigma_\partial (Q))^{-1/2}, $$

hence σ _∂ (P) is invertible if, and only if, σ _∂ (Q) is. We say that $Q\in\varPsi^{m}_{\mathcal{V}}(M_{0};E)$ is fully elliptic if, and only if, P is. In that case, P is Fredholm and Q will also be Fredholm, in the setting of unbounded operators, with $\operatorname {ind}(Q)=\operatorname {ind}(P)$ (see Sect. 4.1).

3.3 The Asymptotically Commutative Case

In this subsection, we prove an index formula for certain classes of pseudodifferential operators on Lie manifolds whose associated groupoids are such that the restrictions at the boundary yield bundles of commutative Lie groups. The main point is giving conditions that yield commutativity of the algebra $\varPsi^{0}(\mathcal{G}_{\partial M})$, using the notation of the previous section, so that Fredholmness depends on invertibility in an algebra of functions, thus reducing to the setting considered in Sect. 2.3. This is known to hold for the scattering and double-edge calculus [41–43, 48, 56]. The dimension of M is denoted by n, as before. Recall that in this section, we do not assume n to be even.

Definition 3.6

Let $(M, \mathcal{W})$ be a connected Lie manifold with Lie algebroid $\pi: A_{\mathcal{W}} \to M$ with the property that any $X \in \mathcal{W}$ vanishes at the boundary ∂M (that is, on any face of the boundary) and the resulting Lie algebras $A_{\mathcal{W}, x} := \pi^{-1}(x)$ are commutative. A Lie manifold $(M, \mathcal{W})$ with this property will be called an asymptotically commutative Lie manifold, and $\mathcal{W}$ will be called commutative at infinity.

(We reserve the notation $\mathcal{W}$ for asymptotically commutative structural Lie algebras of vector fields, whereas $\mathcal{V}$ will denote a general such structural Lie algebra of vector fields.)

Let $(M, \mathcal{W})$ be an asymptotically commutative Lie manifold. Then a groupoid integrating $A_{\mathcal{W}}\vert_{\partial M}$ is $A_{\mathcal{W}}\vert_{\partial M}$ itself (since the commutative Lie algebra ℝⁿ identifies with itself via the exponential map). According to [52], there will be a unique Lie manifold structure on the disjoint union

$$ \mathcal{G}:= (M_0 \times M_0) \cup A_{\mathcal{W}}\vert_{\partial M} $$

(24)

such that $\mathcal{G}$ is a Lie groupoid integrating $A = A_{\mathcal{W}}$. Thus any Lie algebroid associated with an asymptotically commutative Lie manifold has a canonical Lie groupoid integrating it. Let $\overline{A}$ be the sphere bundle obtained by radial compactification of the fibers of A.

Proposition 3.7

Assume $(M, \mathcal{W})$ is an asymptotically commutative Lie manifold and let $\mathcal{G}$ be the canonical Lie algebroid integrating it, as in Eq. (24). Then $\varPsi^{0}(\mathcal{G}_{\partial M})$ is commutative and

$$ \overline{\varPsi^0(\mathcal{G}_{\partial M})} \cong \mathcal{C}(\overline{A}\vert_{\partial M}). $$

Proof

It follows from (24) that the algebra of pseudodifferential operators on $\mathcal{G}_{\partial M}$ coincides with Ψ ⁰(A _∂M ), that is, with the algebra of continuous families of (P _x ) of translation invariant pseudodifferential operators P _x acting on the fibers (A _∂M ) _x of A _∂M , x∈∂M.

Now, the pseudodifferential operators of order zero on a vector space V that are translation invariant coincide with convolution operators with functions whose Fourier transform is in

$$ \widetilde {S^0}(V)=\{p\in \mathcal{C}^\infty(V)\colon p(y,\xi):=p(\xi) \in S^0(T^*V)\} $$

(symbols of order zero that are independent of y). The algebra of convolution operators is commutative, so it follows straightaway that Ψ ⁰(A _∂M ) is commutative. (In particular, the reduced and full C ^∗-norms coincide.) Moreover, one can check that $\overline{\widetilde {S^{0}}(V)}\cong \mathcal{C}(\overline{V})$, with $\overline{V}$ the radial compactification of V. Hence, $\overline{\varPsi^{0} (A_{\partial M})}\cong \mathcal{C}(\overline{A}\vert _{\partial M})$, since there is an isomorphism between elements of Ψ ⁰(A _∂M ) and continuous families in ${\widetilde {S^{0}}(A_{x})}$, which is bounded with respect to the reduced, hence the full, norm. This proves $\overline{\varPsi^{0}(\mathcal{G}_{\partial})} \cong \mathcal{C}(\overline{A}\vert _{\partial M})$, as claimed. □

Note that it follows from the proof that the isomorphism above is really given by the total symbol, as in (4), of the indicial boundary operator. For order m>0 operators, we have $\varPsi^{m}(\mathcal{G}_{\partial M}) = \varPsi^{m}({A}\vert_{\partial M})\cong {\widetilde {S^{m}}({A}\vert_{\partial M})} \subset \mathcal{C}^{\infty}({A}\vert_{\partial M})$, the isomorphism being again given by the total symbol, that is, including the lower-order terms of the symbol. (This total symbol is defined since the resulting operators on the fibers of A→∂M are translation invariant, and hence they are convolution operators. The total symbol is simply the Fourier transform of the resulting convolution distributions.)

As in Sect. 2.3, Eq. (13), consider $\varOmega:= \partial(\overline{A}) = (S^{*}A) \cup\overline{A} \vert_{\partial M}$ such that $\mathcal{C}(\varOmega) = \{(f,g) \in \mathcal{C}(S^{*}{A}) \oplus \mathcal{C}( \overline{A} \vert _{\partial M}), f = g \text{ on } S^{*}{A}_{\partial M} \}$.

Define the boundary symbol for operators on $(M, \mathcal{W})$ by

$$ \sigma_\partial : \overline{\varPsi_{\scriptsize \mathcal{W}}^0(M_0;E)} \to \mathcal{C}(\overline{A}\vert_{\partial M}) $$

(25)

as the map of restriction to the boundary composed with the isomorphism given by the previous proposition. For $P\in\varPsi^{m}_{\mathcal{W}}(M_{0};E)$, the boundary symbol is just given by the total symbol of $\mathcal{R}_{\partial }(Q)=Q\vert_{\partial M}\in\varPsi^{0}(A_{\partial M})$, with $\mathcal{R}_{\partial M}: \varPsi^{m}(\mathcal{G}, r^{*}E)\to\varPsi^{m}(A_{\partial M}, r^{*}E_{\partial M})$ the restriction map and $\pi_{M_{0}}(Q)=P$.

Moreover, it follows from (22) that $\mathcal{C}(\varOmega)$ is the recipient of full symbols of pseudodifferential operators on M, since $\mathcal{C}(\overline{A}\vert_{\partial M})\times_{\mathcal{C}_{0}(S^{*}A_{\partial M})} \mathcal{C}_{0}(S^{*}A) =\mathcal{C}(\partial\overline{A}) = \mathcal{C}(\varOmega)$. We then have a map

$$ \sigma_\mathrm{full} := (\sigma_0, \sigma_\partial ) : \overline{ \varPsi_{\scriptsize \mathcal{W}}^0(M;E)} \to \mathcal{C}(\varOmega), $$

(26)

which is surjective, continuous, and a ∗-algebra morphism. We will see in the next proposition that $\mathcal{K}\subset\ker\sigma_{\mathrm{full}}$, so it follows that $\varOmega=\partial(\overline{A})$ is a comparison space for $\overline{\varPsi_{\scriptsize \mathcal{W}}^{0}(M;E)}$ (see Eq. (14)), and hence the results from Sect. 2.3 apply.

Proposition 3.8

Assume $(M, \mathcal{W})$ is an asymptotically commutative Lie manifold and let $\mathcal{G}$ be the canonical Lie groupoid integrating it, as in Eq. (24). Then $\pi_{M_{0}}$ is injective on $\overline{\varPsi^{0}(\mathcal{G})}$, and hence the following sequence is exact.

(27)

In particular, an operator $P \in\overline{\varPsi^{0}_{\mathcal{W}}(M; E)}$ is Fredholm if, and only if, it is fully elliptic, meaning that $\sigma_{\mathrm{full}}(P) = (\sigma_{0}(P), \sigma_{\partial }(P)) \in \mathcal{C}(\varOmega)$ is invertible.

Proof

The second part will follow from the first part using Proposition 3.4, so we concentrate on proving the injectivity of $\pi_{M_{0}}$. Let I be the kernel of $\pi_{M_{0}}$. We want to show that I={0}. We have that $\pi_{M_{0}}$ is injective on the subalgebra of compact operators of $\varPsi^{0}_{\mathcal{W}}(M,E)$, so $I \cap \mathcal{K}= 0$. It follows that (σ ₀,σ _∂M ) is injective on I, since it has kernel $\mathcal{K}$.

Let $P \in I \subset\varPsi^{0}_{\mathcal{W}}(M,E)$. We can recover the principal symbol of P from its action on M ₀ [3, 38, 43], so we can assume $P \in I \subset\varPsi^{m}_{\mathcal{W}}(M,E)$, with m<0. Next, we replace P with (P ^∗ P)^k, which, for k large, has arbitrarily small order. We can therefore assume that m<−n. (This is the usual “descent” argument in Hörmander’s trick.) The Fourier transform (as in Eq. (18)) then allows us to recover the boundary symbol of P since for x approaching the boundary, the exponential map increases its radius of injectivity (so the cutoff χ will affect less and less the kernel of the resulting operator). This shows that $\pi_{M_{0}}$ is injective on $\overline{\varPsi^{0}(\mathcal{G})}$. □

In this case, let [σ ₀(P)]∈K ⁰(TM ₀) denote the K ⁰-theory class associated with P, as in (16), and $[\sigma_{\mathrm{full}}(P)]:=[(\sigma_{\partial }(P),\sigma_{0}(P))]\in K_{1}(\mathcal{C}(\varOmega))\cong K^{1}(\varOmega)$ denote the class in K ¹. As before, let Td(T _ℂ M) denote the Todd class of the complexified tangent bundle of M, and $\pi: \overline{T^{*}M}\to M$. Also, we denote by $\pi_{\varOmega}: \varOmega= \partial(\overline{A}) \to M$ the natural projection. From Theorem 2.8 it finally follows:

Theorem 3.9

Let $(M,\mathcal{W})$ be an asymptotically commutative Lie manifold with Lie algebroid A, $\varOmega:= \partial(\overline{A})$, and let $P \in\overline{\varPsi_{\mathcal{W}}^{0}(M;E)}$ be an elliptic operator with σ _∂ (P) invertible in $\mathcal{C}(\overline{A}_{\partial M})$. Then,

where $[\widetilde {\sigma}_{\mathrm{full}}(P)] \in K^{0}(TM_{0})$ is defined using Lemma 2.7.

Our main example of an asymptotically commutative Lie manifold $(M, \mathcal{W})$ is obtained as follows. Let $(M, \mathcal{V})$ be a Lie manifold and let x _k be boundary defining functions of the hyperfaces of M. Choose a _k ∈ℕ={1,2,…}. Then, as in Eq. (3), we introduce

$$ \mathcal{W}:= f \mathcal{V}, \quad\mbox{with } f:= \prod x_k^{a_k}. $$

Then $\mathcal{W}$ is also a structural Lie algebra of vector fields, since it is closed for Lie brackets, and a finitely generated, projective $\mathcal{C}^{\infty}(M)$-module. Hence $(M,\mathcal{W})$ is a Lie manifold that is easily seen to be asymptotically commutative.

The previous result extends the known index formulas for the scattering calculus on manifolds with boundary, where $\mathcal{V}_{sc}:= x\mathcal{V}_{b}$, x is a boundary defining function and $\mathcal{V}_{b}$ is the Lie algebra of vector fields tangent to the boundary, and for the double-edge calculus, where $\mathcal{V}_{de}=x\mathcal{V}_{e}$, with $\mathcal{V}_{e}$ the edge vector fields induced by a fibration of the boundary [36, 40, 42, 48]. Moreover, the index formula above can be proved in the same way considering families of pseudodifferential operators over a compact base space B (the index now takes values in K ⁰(B)) using a generalization of the Atiyah–Singer index theorem for families of asymptotically multiplication operators. In this sense, Theorem 3.9 yields the result in [40] for families of scattering pseudodifferential operators.

In the next section, we will apply the index formula above to compute the index of perturbed Dirac operators on general Lie manifolds.

4 Perturbed Dirac Operators

Throughout this section, we let M ₀ be a non-compact, even dimensional manifold M ₀, which, as before, is assumed to be the interior of a Lie manifold $(M, \mathcal{V})$. We fix a set {x _k } of defining functions of M and let

$$ f: = \prod x_k^{a_k},\quad a_k \in \mathbb {N}, $$

(28)

(so a _k >0). In this section, we consider a Dirac operator $\hspace* {.5mm}/\hspace* {-2.35mm}D$ coupled with a potential V, that is, an operator of the form

$$ T = \hspace* {.5mm}/\hspace* {-2.35mm}D+ V := \hspace* {.5mm}/\hspace* {-2.35mm}D\widehat{\otimes}1 + 1 \widehat{\otimes}V $$

(29)

on compactly supported sections of some vector bundles defined on M ₀. By a potential we shall always mean an odd, self-adjoint endomorphism of a ℤ₂-graded vector bundle E over M ₀ and endowed with a compatible connection. An operator T of this type will be called a Callias-type operator. (More precisely, T is the closure of $T \widehat{\otimes}1 + 1 \widehat{\otimes}V$.) We assume the potential V to be of the form

$$ V := f^{-1} V_0 = \prod x_k^{-a_k} V_0, $$

where V ₀ extends to a smooth endomorphism of E on M, invertible at the boundary. In particular, the potential V is unbounded.

We apply the results of the previous section to give a cohomological formula for the index of $T^{+} := (\hspace* {.5mm}/\hspace* {-2.35mm}D+ V)^{+}$ on natural Sobolev spaces. The main point is to reduce the calculation of the index of T ⁺ to the case of a Dirac operator coupled with a bounded potential on the asymptotically commutative Lie manifold $(M, \mathcal{W})$ defined by $\mathcal{W}:= f\mathcal{V}$, and show that the index can be obtained from Theorem 3.9. More precisely, we shall show that

$$ \operatorname {ind}(T^+) = \operatorname {ind}(Q^+) \quad\text{for}\ Q := f^{1/2} T^+ f^{1/2} \in\varPsi^1_{\mathcal{W}}(M; F_0, F_1 ), \ \mathcal{W}:= f \mathcal{V}, $$

(30)

for suitable vector bundles F ₀ and F ₁. We then use that

$$ P := Q(1 + Q^*Q)^{-1/2} \in\overline{\varPsi^0_{\mathcal{W}}(M; F_0, F_1)} $$

also satisfies $\operatorname {ind}(P^{+}) = \operatorname {ind}(Q^{+})$. Finally, we show that $\operatorname {ind}(P^{+})$, and hence also $\operatorname {ind}(T^{+}) = \operatorname {ind}(P^{+})$, can be computed using Theorem 3.9.

See also the results of Albin–Rochon [1] and Moroianu [50] for results on the decay of the functions in the null spaces of T, Q, and Q ^∗, which also give a heuristic explanation for our coincidence of the indices of P ⁺ and Q ⁺.

4.1 Dirac and Callias Operators

Let W and E be ℤ₂-graded, Hermitian vector bundles over M and assume that both W and E are endowed with compatible connections. We also endow the tensor product bundle W⊗E over M with the usual grading and denote by $W \widehat {\otimes}E$ the resulting ℤ₂-graded vector bundle, with the tensor product connection, namely,

$$ (W\widehat {\otimes}E)^+=( W^+\otimes E^+) \oplus(W^-\otimes E^-) $$

and

$$ (W\widehat {\otimes}E)^-= (W^-\otimes E^+) \oplus(W^+\otimes E^-). $$

If $V \in \operatorname {End}(E)$ is an endomorphism, then it acts on C ^∞(E) as a (pseudo)differential operator of order 0.

Let $D\in\varPsi^{m}_{\mathcal{V}}(M,W)$. Then we define $D\widehat{\otimes}1$ acting on $\mathcal{C}^{\infty}(W\widehat {\otimes} E)$ as in [40]. (Note that, since D is not C ^∞(M)-linear, extra conditions might be needed to assure that $D \widehat {\otimes} 1$ is well defined as the tensor product of two endomorphisms, for instance, that the connection on E be flat.) For D a Dirac operator—which is really the only case we need to obtain our main result—this comes for free, since one recovers the usual definition of $D \widehat{\otimes}1$ as the Dirac operator on the tensor product bundle endowed with the tensor product connection.

Definition 4.1

An operator $T: \mathcal{C}^{\infty}_{c}(M_{0}; W\widehat {\otimes} E) \to \mathcal{C}^{\infty}_{c}(M_{0}; W\widehat {\otimes} E)$ is said to be a Callias-type pseudodifferential operator on the Lie manifold $(M,\mathcal{V})$ if

$$ T := D + V := D \widehat {\otimes} 1 + 1\widehat {\otimes} V = \left( \begin{array}{c@{\quad}c} D\otimes1 & -1 \otimes V \\ 1 \otimes V & D\otimes1 \end{array} \right), $$

where $D\in\varPsi^{m}_{\mathcal{V}}(M,W)$, m>0, is an odd, symmetric elliptic operator and $V\in \operatorname {End}(E{\vert_{M_{0}}})$ is odd and self-adjoint. We shall also assume V to be invertible outside a compact subset of M ₀. We refer to V as a potential. The closure of an operator of the form T=D+V will also be called a Callias-type operator.

When $D \widehat {\otimes} 1$ is the (generalized) Dirac operator associated with a given Clifford bundle structure and connection on $W\widehat { \otimes} E$, these operators are also called Dirac–Schrödinger operators, and were first considered by Callias, in the odd-dimensional Euclidean space [18]. See also [15, 17, 30, 31] and references therein for more results on index theory of Dirac–Schrödinger and Callias-type operators on even-dimensional manifolds. See also [16].

Remark 4.2

On odd-dimensional manifolds, the Callias-type operators are of the form $\hspace* {.5mm}/\hspace* {-2.35mm}D+ iV$, where V is self-adjoint and invertible at infinity. See [5, 6, 13, 14, 18, 29, 40, 59] for more on the index of Callias-type operators in the odd case.

Recall that a symmetric (hence closable) operator T is essentially self-adjoint if its closure is self-adjoint, that is, if 〈Tx,y〉=〈x,Ty〉, for all $x,y \in \mathcal{D}(T)=\mathcal{D}(T^{*})$. (We shall always denote the minimal closure of an operator by the same letter.)

In the following lemma, we assume that the potential V ₀ extends to an endomorphism of E over M; in particular, it is bounded. (We will prove such a result for an unbounded potential in Sect. 4.4.)

Lemma 4.3

Let $D \in\varPsi^{m}_{\mathcal{V}}(M,W)$ be an odd, symmetric, elliptic operator. Assume that V ₀ extends to an endomorphism of E over M, as before. Then the Callias-type operator $T =D + V_{0} \in\varPsi^{m}_{\mathcal{V}}(M_{0}; W \otimes E)$ is elliptic and essentially self-adjoint on $\mathcal {C}^{\infty}_{c}(M_{0}; W\otimes E)$.

Proof

Ellipticity follows from $\sigma_{m}(T) = \sigma_{m}(D)\widehat {\otimes } 1$. The fact that T is essentially self-adjoint follows, for instance, from [43] (Theorem 7.1), which states that a (possibly unbounded) symmetric, elliptic operator in $\varPsi^{m}_{\mathcal{V}}(M; W\otimes E)$, with m>0, is essentially self-adjoint, where we have identified $\varPsi^{m}_{\mathcal{V}}(M; W\otimes E) = \pi_{M_{0}}(\varPsi^{1}(\mathcal{G}; r^{*}(W\otimes E)))$, as in Theorem 3.3. □

We shall work with unbounded Fredholm operators. It will then be useful to recall the way they are introduced. Let T be a possibly unbounded operator with domain $\mathcal{D}(T)$ and codomain H. We shall always replace T by its closure, so assume T is closed and endow $\mathcal{D}(T)$ with the graph norm. Then T is Fredholm if, by definition, the induced bounded operator $T : \mathcal{D}(T) \to H$ is Fredholm (in the usual sense of having finite-dimensional kernel and cokernel). In particular, a pseudodifferential operator T ₁ acting between sections of E ₀ with range sections of E ₁ is Fredholm if, and only if, $T_{2} := T_{1}(1+T_{1}^{*}T_{1})^{-1/2}$ is a Fredholm operator and, in this case, $\operatorname {ind}(T_{1})=\operatorname {ind}(T_{2})$.

We are interested in computing the index of

$$ T^+ =(D + V)^+ : \mathcal{C}^\infty_c(M_0; (W \widehat {\otimes} E)^+) \to \mathcal{C}^\infty_c(M_0; (W\widehat {\otimes} E)^-), $$

(31)

which we shall prove to be Fredholm between suitable Sobolev spaces. Note that, with respect to the grading, we can write

$$ T^+ = \left( \begin{array}{c@{\quad}c} D^+\otimes1 & -1 \otimes V^- \\ 1 \otimes V^+ & D^-\otimes1 \end{array} \right). $$

(32)

Most of our results work for general odd, elliptic, positive pseudodifferential operators $D \in\varPsi^{m}_{\mathcal{V}}(M; W)$. However, for simplicity and because this is the most useful case in applications, we shall mainly be interested in the case when D is a generalized Dirac operator. Recall that, in any case, the Dirac operators generate all classes in K-homology, so we can always assume D to be a Dirac operator.

4.2 Dirac Operators on Lie Manifolds

We introduce here generalized Dirac operators on Lie manifolds following [4]. Let $(M, \mathcal{V})$ be an even-dimensional Lie manifold endowed with a compatible metric g on M ₀ and let W be a Clifford module over M endowed with an A ^∗-valued connection ∇^W and a Clifford multiplication bundle map c:A⊗W→W. (Recall that a compatible metric on M ₀ is a metric on TM ₀ that extends to A→M.) The restrictions of W, c, and ∇^W to M ₀ reduce to the classical notions of a Clifford bundle together with an admissible connection [35, 44, 48].

Definition 4.4

The generalized Dirac operator $\hspace* {.5mm}/\hspace* {-2.35mm}D: \mathcal{C}^{\infty}(M;W) \to \mathcal{C}^{\infty}(M;W)$ associated with W is then defined as the composition

(33)

where ϕ:A ^∗→A is the isomorphism given by the metric.

Since both the Clifford multiplication c and the A ^∗-valued connection are $\mathcal{V}$-differential operators, of order 0 and 1, respectively, we have that $\hspace* {.5mm}/\hspace* {-2.35mm}D\in \operatorname {Diff}_{\mathcal{V}}^{1}(M;W)$. The principal symbol $\sigma_{1}(\hspace* {.5mm}/\hspace* {-2.35mm}D)\xi= ic(\xi)\in \operatorname {End}(W)$ is invertible for any ξ≠0, and hence $\hspace* {.5mm}/\hspace* {-2.35mm}D$ is elliptic. It follows from classical results that $\hspace* {.5mm}/\hspace* {-2.35mm}D$ with domain $\mathcal{C}^{\infty}_{c}(M_{0}; W) \subset L^{2}(M_{0}; W)$ is essentially self-adjoint (i.e., its closure is self-adjoint), since M ₀ is complete.

We can also define Dirac operators on groupoids: If $\mathcal{G}$ is a d-connected groupoid integrating $A = A(\mathcal{V})$, then we can consider the Clifford module r ^∗ W and endow $\mathcal{G}$ with an admissible connection $\nabla^{\mathcal{G}}\in \operatorname {Diff}(\mathcal{G}; W, W\otimes A^{*})$ such that $\pi_{M_{0}}(\hspace* {.5mm}/\hspace* {-2.35mm}D^{\mathcal{G}})=\hspace* {.5mm}/\hspace* {-2.35mm}D$, where $\hspace* {.5mm}/\hspace* {-2.35mm}D^{\mathcal{G}}$ is the associated Dirac operator on $\mathcal{G}$ (see [43] for details).

Recall that M is even-dimensional and assume that W is ℤ₂-graded, with the grading given by the chirality operator. Also, let E be a Hermitian ℤ₂-graded vector bundle over M and $V\in \operatorname {End}(E)$ a potential (so odd, self-adjoint, and invertible outside a compact subset of M ₀). We assume as before that W and E are endowed with compatible connections. Note that in this case, $\hspace* {.5mm}/\hspace* {-2.35mm}D\widehat{\otimes}1$ can be defined directly as $e(\hspace* {.5mm}/\hspace* {-2.35mm}D\oplus\cdots \oplus \hspace* {.5mm}/\hspace* {-2.35mm}D) e$, where e is the projection corresponding to an embedding of E in a trivial bundle and $\hspace* {.5mm}/\hspace* {-2.35mm}D\oplus\cdots\oplus \hspace* {.5mm}/\hspace* {-2.35mm}D$ coincides with the Dirac operator on W⊗ℂⁿ, $n= \operatorname {rk}(E)$. In Sect. 4.4, we shall be computing the index of the chiral operator

$$T^+ =(\hspace* {.5mm}/\hspace* {-2.35mm}D+ V)^+=( \hspace* {.5mm}/\hspace* {-2.35mm}D\widehat {\otimes} 1 + 1\widehat {\otimes} V)^+ , $$

as in (31), with an unbounded potential V, on natural Sobolev spaces on the Lie manifold $(M, \mathcal{V})$.

4.3 The Case of Bounded Potentials

Let $(M, \mathcal{W})$ be an asymptotically commutative Lie manifold. Recall that we assume that n, the dimension of M, is even. In this subsection, we consider Callias-type operators

$$ Q := D + V_0 = D \widehat {\otimes} 1 + 1\widehat {\otimes} V_0 \quad\text{with } D \in\varPsi^m_{\mathcal{W}}(M_0; W\otimes E), $$

(34)

where D is an elliptic, symmetric, odd pseudodifferential operator, as in Definition 4.1. (Note that, in any case, to obtain our main result, we will only need the case when $D=\hspace* {.5mm}/\hspace* {-2.35mm}D$, a Dirac operator, as in the previous section). We assume V ₀ to be a bounded potential on M ₀ that extends to a smooth endomorphism of E over M that is invertible on ∂M (so, in particular, V ₀ is odd and symmetric). It follows from Lemma 4.3 that Q is elliptic and essentially self-adjoint on $\mathcal {C}^{\infty}_{c}(M_{0})$. We shall give a K-theory formula for the index of Q ⁺.

We define the total symbol K-theory classes σ _full(Q)∈K ¹(Ω) and $\widetilde {\sigma}_{\mathrm{full}}(Q) \in K^{0}(TM_{0})$ of Q in a way similar to the case of order zero symbols. First, recall that the boundary symbol $\sigma_{\partial }: \varPsi_{\scriptsize \mathcal{W}}^{m}(M_{0};E) \to \mathcal{C}({A}\vert_{\partial M})$ is given by

$$ \sigma_\partial (P) = \sigma_m^\mathrm{tot}(\mathcal{R}_\partial (S)) = \sigma_m^\mathrm{tot}(S\vert_{\partial M}), $$

with $\pi_{M_{0}}(S)=P$, $\mathcal{R}_{\partial }: \varPsi^{m}(\mathcal{G})\to\varPsi^{m}(\mathcal{G}_{\partial M})$ is restriction to the boundary and $\sigma_{m}^{\mathrm{tot}}(S_{x})$, x∈∂M, is the total symbol of the operator S _x on A _x (including lower-order terms).

Lemma 4.5

Let Q be as in Eq. (34) and P:=Q(1+Q ²)^−1/2. Then $P \in\overline{\varPsi^{0}_{\mathcal{W}}(M_{0}; W \otimes E)}$ is fully elliptic, in the sense that its principal symbol σ ₀(P) and the boundary symbol σ _∂ (P), defined by continuity, are invertible.

Proof

It follows from Lemma 3.5 that $P \in\overline{\varPsi^{0}_{\mathcal{W}}(M_{0}; W \otimes E)}$. We have that P is elliptic, since Q is. To understand the boundary operators, since $\mathcal{W}$ is commutative at the boundary, hence $\mathcal{G}_{\partial M} = A_{\partial M}$ is amenable, we only need invertibility on fibers $\mathcal{G}_{x}=A_{x}$, x∈∂M (see the remark after Theorem 3.4). Let $S\in\varPsi^{m}(\mathcal{G}; r^{*}W)$ be such that $\pi_{M_{0}}(S)=D$. Therefore, we need to look at the symbol of the operators S _x coupled with the constant potential V ₀(x) acting on the fiber A _x , for each x∈∂M. The invertibility of the boundary indicial operator $\sigma_{\partial }(D + V_{0})_{x}(\xi) = \sigma^{\mathrm{tot}}(S_{x}) \widehat {\otimes}1 + 1\widehat {\otimes} V_{0}(x)$, ξ∈A _x then follows from the fact that V ₀(x) is invertible for each x∈∂M (noting that $\alpha \widehat {\otimes}1 + 1\widehat {\otimes} \beta\in \operatorname {End}(W_{x}\widehat{\otimes}E_{x})$ is invertible if α or β are invertible.) □

Note that when $D=\hspace* {.5mm}/\hspace* {-2.35mm}D$ is a Dirac operator on $(M, \mathcal{W})$ then, using the notation as above, $S_{x}=\hspace* {.5mm}/\hspace* {-2.35mm}D_{x}$ is a Dirac operator on A _x and $\sigma_{\partial }(\hspace* {.5mm}/\hspace* {-2.35mm}D+ V)_{x}(\xi)= ic(\xi) \widehat {\otimes} 1 + 1 \widehat {\otimes} V_{0}(x)$. (This is due to the fact that the restriction of a Dirac operator to the boundary is again a Dirac operator [43].)

The following lemma provides the definitions of the total symbol K-theory classes σ _full(Q)∈K ¹(Ω) and $\widetilde {\sigma}_{\mathrm{full}}(Q) \in K^{0}(TM_{0})$. Let us introduce the K-theory class [V ₀] defined by the endomorphism V ₀ as usual [7, 39]

$$ [V_0]:= [E^+, E^-, V_0] \in K^0(M_0) = K^0(M;\partial M)\subset K^0(M). $$

Recall that $\varOmega:= \partial(\overline{A}) = (S^{*}A) \cup\overline{A} \vert_{\partial M}$ (as in Sect. 3.3).

Lemma 4.6

Let Q and P be as in Lemma 4.5 and define [σ _full(Q)]:=[σ _full(P)]∈K ¹(Ω) and $[\widetilde {\sigma}_{\mathrm{full}}(Q)] := [\widetilde {\sigma}_{\mathrm{full}}(P)] \in K^{0}(TM_{0})$. Then

$$ \partial [\sigma_\mathrm{full}(Q)] = [\widetilde {\sigma}_\mathrm{full}(Q)] $$

and $[\widetilde {\sigma}_{\mathrm{full}}(Q)]$ can be represented by the endomorphism $\sigma_{m}(D) \widehat{\otimes}1 + 1 \widehat{\otimes}V_{0}$. In particular,

$$ [\widetilde {\sigma}_\mathrm{full}(Q)] = [\sigma_m(D)] \otimes\pi^*[V_0], $$

where [σ _m (D)]∈K ⁰(TM) and [V ₀]∈K ⁰(M,∂M) are the classes defined by the corresponding morphisms and π:TM→M is the natural projection.

Proof

The relation $\partial [\sigma_{\mathrm{full}}(Q)] = [\widetilde {\sigma}_{\mathrm{full}}(Q)]$ follows from definitions and from Lemma 2.7. Let us choose a smooth function σ _m ∈S ^m(A ^∗) such that σ _m represents σ _m (D) and on A ^∗| _∂M it is equal to the total symbol of D. Then let $p = \sigma_{m} \widehat{\otimes}1 + 1 \widehat{\otimes}V_{0} \in \mathcal {C}^{\infty}(A^{*}) = \mathcal {C}^{\infty}(A)$, where we have used a fixed metric on A to identify A with A ^∗, as before. From the definition of P in terms of Q and from the properties of the principal symbol, we have that

On the other hand, at the boundary, we have

$$ \sigma_\partial (P) = \sigma_\partial (Q)/\sqrt{1 + \sigma_\partial (Q)^2} = p/\sqrt{1 + p^2}. $$

Therefore, $\sigma_{\mathrm{full}}(P) = p/\sqrt{1 + p^{2}}$ on Ω. Hence the K-theory class $[\widetilde {\sigma}_{\mathrm{full}}(P)]$ is obtained from the endomorphism $p/\sqrt{1 + p^{2}}$ defined on TM ₀, which obviously extends $p/\sqrt{1 + p^{2}}$ from Ω to the whole of $\overline{A} \supset TM_{0}$. We obtain that the endomorphism p, and hence also $\sigma_{m}(D) \widehat{\otimes}1 + 1 \widehat{\otimes}V_{0}$, represents $[\widetilde {\sigma}_{\mathrm{full}}(Q)]$.

From the definition of the tensor product in K-theory, we have that

$$ [\sigma_{m}(D)] \otimes\pi^*[V_0] = [\pi^*(W\widehat{\otimes}E)^+, \pi^*(W\widehat{\otimes}E)^-, \sigma_{m}(D)\widehat{\otimes}1 \oplus 1 \widehat{\otimes}V_0], $$

where

$$ \sigma_{m}(D)\widehat{\otimes}1 \oplus1\widehat{\otimes}V_0 = \left( \begin{array}{c@{\quad}c} \sigma_m(D^+)\otimes1 & -1 \otimes V_0^- \\ 1 \otimes V_0^+ & \sigma_m(D^-)\otimes 1 \end{array} \right). $$

It follows that the K-theory class $[\widetilde {\sigma}_{\mathrm{full}}(Q)]$, where $Q=D \widehat {\otimes}1 + 1 \widehat {\otimes} V_{0}$, is represented by the same morphism as [σ _m (D)]⊗π ^∗[V ₀]. So these two classes are equal. □

We shall need the Sobolev spaces $H_{\mathcal{W}}^{m}(M_{0})$ defined by $\mathcal{W}$ (more precisely by the metric determined by $\mathcal{W}$ [2, 3]).

$$ H_{\mathcal{W}}^m(M_0) := \{u\in L^2(M), Du\in L^2(M_0) \text{ for all } D\in \operatorname {Diff}^m_\mathcal{W}(M_0)\}. $$

(35)

The space $H_{\mathcal{W}}^{m}(M_{0})$ is the domain of any elliptic pseudodifferential operator in $\varPsi_{\mathcal{W}}^{m}(M_{0})$, m>0, acting on L ²(M ₀). For m<0 we use duality.

We now show that $\operatorname {ind}(Q^{+})$ can be computed using Theorem 3.9.

Theorem 4.7

Let Q=D+V ₀ be a Callias-type pseudodifferential operator with a bounded potential V ₀ as in Lemmas 4.5 and 4.6. In particular, we assume that V ₀ is a smooth potential on M that is invertible on ∂M. Then $Q^{+} : H_{\mathcal{W}}^{m}(M_{0})\to L^{2}(M_{0})$ is Fredholm and

Proof

Let P:=Q(1+Q ^∗ Q)^−1/2, as before. Then $P \in\overline{\varPsi^{0}_{\mathcal{W}}(M_{0}; W \otimes E)}$ is fully elliptic, by the previous lemma (Lemma 4.5). By Proposition 3.4, it follows that P is Fredholm on L ²(M ₀) and hence Q is also Fredholm as an operator $H_{\mathcal{W}}^{m}(M_{0})\to L^{2}(M_{0})$. Moreover, we now have

by Theorem 3.9 applied to P ⁺ and Lemma 4.6. □

We are mainly interested in the case when

$$ Q = \hspace* {.5mm}/\hspace* {-2.35mm}D+ V_0 := \hspace* {.5mm}/\hspace* {-2.35mm}D\widehat {\otimes} 1 + 1 \widehat {\otimes} V_0, $$

(36)

where $\hspace* {.5mm}/\hspace* {-2.35mm}D$ is a Dirac operator acting on the sections of some Clifford bundle W. As before, we assume V ₀ is a potential (i.e., an odd, self-adjoint endomorphism of a ℤ₂-graded bundle E) that is invertible outside a compact subset of M ₀ such that V ₀ extends smoothly to M, to be invertible at ∂M. In particular, $Q\in\varPsi^{1}_{\mathcal{W}}(M_{0}; W \otimes E)$.

To get an even more explicit formula for the index of the coupled Dirac operators $\hspace* {.5mm}/\hspace* {-2.35mm}D+ V_{0}$, let us assume now that M has a spin^c-structure, with canonical spin^c-bundle S and associated Dirac operator $\hspace* {.5mm}/\hspace* {-2.35mm}D_{S}$. In particular, M is oriented, and we let [M]∈H _n (M,∂M) denote its fundamental class. Let $W = S \widehat{\otimes}F$, with F a complex vector bundle over M. Then $\hspace* {.5mm}/\hspace* {-2.35mm}D_{F} := \hspace* {.5mm}/\hspace* {-2.35mm}D_{S} \otimes F$ is the Dirac operator twisted with F.

Corollary 4.8

Let $\hspace* {.5mm}/\hspace* {-2.35mm}D_{F}$ be the Dirac operator twisted with F and $Q = \hspace* {.5mm}/\hspace* {-2.35mm}D_{F} + V_{0}$ be the perturbed twisted Dirac operator associated with V ₀, where V ₀ is a bounded potential invertible at ∂M on an asymptotically commutative spin ^c Lie manifold $(M, \mathcal{W})$. Then Q ⁺ is Fredholm and

$$ \operatorname {ind}(Q^+) = {\widehat{A}(M) \operatorname {ch}_0 ([F \otimes V_0] ) [M]}. $$

Proof

It is known classically that

$$ p_! \operatorname {ch}_0(\sigma(\hspace* {.5mm}/\hspace* {-2.35mm}D_F^+)) Td(T_\mathbb {C}M) =\widehat{A}(M) \operatorname {ch}_0[F], $$

where p _! is integration over the fiber and $\widehat{A}(M)\in H^{*}(M)$ is the $\widehat {A}$-genus of M (see [44]). The result then follows right away from Theorem 4.7. □

4.4 The Case of Unbounded Potentials

In this subsection, we are back to a general, even-dimensional, Lie manifold $(M, \mathcal{V})$. Let

$$ T := \hspace* {.5mm}/\hspace* {-2.35mm}D+ V := \hspace* {.5mm}/\hspace* {-2.35mm}D\widehat {\otimes} 1 + 1 \widehat {\otimes} V, $$

(37)

where $\hspace* {.5mm}/\hspace* {-2.35mm}D\in \operatorname {Diff}^{1}_{\mathcal{V}}(M_{0}; W)$ is a Dirac operator associated with $\mathcal{V}$, and, as before, both W and E are endowed with compatible connections (so in particular, $\hspace* {.5mm}/\hspace* {-2.35mm}D\widehat {\otimes} 1$ is well defined). We will consider here unbounded potentials, in that we assume moreover that, on M ₀,

$$ V = f^{-1}V_0, \quad f:=\prod x_k^{a_k}, $$

(38)

where V ₀ is bounded and it extends to a smooth endomorphism of E over M that is invertible on ∂M (at infinity) and x _k are boundary defining functions of the hyperfaces of M with a _k ∈ℕ={1,2,…}. This section contains the hard analysis needed for our main result.

Our first goal is to show that T is essentially self-adjoint with domain a suitable weighted Sobolev space. We want to prove a formula for the index of $T^{+} = (\hspace* {.5mm}/\hspace* {-2.35mm}D+ V)^{+}$. Our strategy is to reduce this problem to a question on operators with bounded potential by writing

$$ T = f^{-1/2}Qf^{-1/2}, \quad\mbox{ with}\ Q:=f^{1/2}\hspace* {.5mm}/\hspace* {-2.35mm}Df^{1/2} + V_0. $$

(39)

In fact, let $\mathcal{W}:= f\mathcal{V}$ and let g be the given metric compatible with $\mathcal{V}$. Then g ₀:=f ⁻² g is a metric compatible with $\mathcal{W}$ and hence

$$ \hspace* {.5mm}/\hspace* {-2.35mm}D_\mathcal{W}:= f^{1/2}\hspace* {.5mm}/\hspace* {-2.35mm}Df^{1/2} $$

is the Dirac operator associated with the Lie manifold structure defined by $\mathcal{W}$ and metric g ₀ [10, 11, 37, 44, 51]. Actually, to identify $\hspace* {.5mm}/\hspace* {-2.35mm}D_{\mathcal{W}}$ with $f^{1/2}\hspace* {.5mm}/\hspace* {-2.35mm}Df^{1/2}$, we need to rescale the volume forms also, a fact that we ignore throughout, in order to simplify the notation.

We then have that $Q \in{ \operatorname {Diff}^{1}_{\scriptsize \mathcal{W}}}(M; W\otimes E)$ is a Callias-type Dirac operator on $(M,\mathcal{W})$ with a bounded, invertible potential. In particular, it is elliptic and essentially self-adjoint on $\mathcal{C}_{c}^{\infty}(M_{0})$. As before, we still denote its self-adjoint closure by Q.

We now define weighted Sobolev spaces defined by $\mathcal{W}$

$$ \mathcal{K}_a^m(M_0) : = f^{a}H_{\mathcal{W}}^m(M_0), $$

(40)

where a∈ℝ and $f = \prod x_{k}^{a_{k}}$, as before. If E→M is a smooth vector bundle, then the spaces $\mathcal{K}_{a}^{m}(M_{0}; E)$ are defined similarly. We remark that all the weighted Sobolev spaces used below are with respect to $\mathcal{W}$. (One can check that $\mathcal{K}_{a}^{m}(M_{0}; E; \mathcal{V})=\mathcal{K}_{a-n/2}^{m}(M_{0}; E;\mathcal{W})$.) We have the following elliptic regularity result from [2].

Theorem 4.9

Assume that $Q_{0} \in\varPsi_{\mathcal{W}}^{k}(M_{0};E)$ is elliptic and $h \in \mathcal{K}_{a}^{s}(M_{0}; E)$ is such that $Q_{0}h \in \mathcal{K}_{a}^{m-k}(M_{0}; E)$. Then $h \in \mathcal{K}_{a}^{m}(M_{0}; E)$.

Applying this result to $Q = f^{1/2} T f^{1/2}=\hspace* {.5mm}/\hspace* {-2.35mm}D_{\mathcal{W}}+ V_{0}$, we obtain the following.

Lemma 4.10

Let $h \in \mathcal{K}_{a}^{s}(M_{0}; E)$ be such that $Th \in \mathcal{K}_{a-1}^{m-1}(M_{0}; E)$. Then $h \in \mathcal{K}_{a}^{m}(M_{0}; E)$.

We shall also need the following lemma. Before, we remark that it follows from the definitions that multiplication by f ^s defines an isomorphism $f^{s}: \mathcal{K}_{a}^{m}(M_{0}; E_{0}) \to \mathcal{K}_{a+s}^{m}(M_{0};E_{0})$, for any a,s. In particular, if $P\in\varPsi_{\mathcal{W}}^{k}(M_{0})$, with $P: H^{m}_{\mathcal{W}}(M_{0};E_{0})\to H^{m-k}_{\mathcal{W}}(M_{0};E_{0})$, then $f^{s}Pf^{-s}: \mathcal{K}^{m}_{s}(M_{0}; E_{0})\to \mathcal{K}_{s}^{m-k}(M_{0}; E_{0})$ is Fredholm if, and only if, P is. Moreover, it is known that $f^{s}\varPsi_{\mathcal{W}}^{k}(M_{0};E_{0})f^{-s} = \varPsi_{\mathcal{W}}^{k}(M_{0};E_{0})$ (Proposition 4.3 [2]), so any such P is also defined as an operator, still denoted by P,

$$P: \mathcal{K}^m_s(M_0;E_0) \to \mathcal{K}^{m-k}_s(M_0;E_0), $$

for any s.

Lemma 4.11

Let $Q_{0} \in\varPsi_{\mathcal{W}}^{k}(M_{0}; E_{0})$, k∈ℤ₊, be fully elliptic. Then

$$ Q_{a,b,c} := f^b Q_{{0}} f^{c} : \mathcal{K}_a^m(M_0; E_0) \to \mathcal{K}_{a+b+c}^{m-k}(M_0; E_0) $$

is Fredholm and its index is independent of m, a, b, and c, in the sense that

$$ \operatorname {ind}(Q_{a, b, c}) = \operatorname {ind}(Q_{0,0,0}). $$

Proof

Let us notice first that Q _0,0,0 is Fredholm due to Proposition 3.8 (since we assumed Q ₀ to be fully elliptic). It follows that $f^{s} Q_{0,0,0} f^{-s}: \mathcal{K}^{m}_{s}(M_{0}; E_{0}) \to \mathcal{K}_{s}^{m-k}(M_{0}; E_{0})$ is also Fredholm and $\operatorname {ind}(f^{s} Q_{0,0,0} f^{-s}) = \operatorname {ind}(Q_{0,0,0})$. Note also that Q _a,b,c is indeed well defined, by the remarks above. Write Q _a =Q _a,0,0.

Next, we notice that $f^{s} P f^{-s} - P = f^{s}(P f^{-s} - f^{-s}P) \in f \varPsi_{\mathcal{W}}^{k-1}(M_{0}; E_{0})$ for any $P\in \varPsi_{\mathcal{W}}^{k}(M_{0}; E_{0})$ by the specific form of the Lie algebra of vector fields $\mathcal{W}= f\mathcal{V}$. Moreover, $f^{s} Q_{{a-s}} f^{-s} - Q_{{a}} : \mathcal{K}_{a}^{m}(M_{0}; E_{0}) \to \mathcal{K}_{a}^{m-k}(M_{0}; E_{0})$ is compact, since $f \mathcal{K}_{a}^{m}(M_{0}; E_{0}) \to \mathcal{K}_{a}^{m-k}(M_{0}; E_{0})$ is compact by [3]. With s=a, we conclude that Q _a is Fredholm and also that Q _a and f ^s Q _a−s f ^−s have the same index for any s.

If follows that the index of $Q_{a} : \mathcal{K}_{a}^{m}(M_{0}; E_{0}) \to \mathcal{K}_{a}^{m-k}(M_{0}; E_{0})$ is independent of a. Using this with a replaced by a+c and using the fact that $f^{s} : \mathcal{K}_{a}^{m}(M_{0}; E_{0}) \to \mathcal{K}_{a+s}^{m}(M_{0}; E_{0})$ is an isomorphism, we obtain the desired result. □

We shall use this lemma to prove the following crucial result.

Proposition 4.12

The operators

$$ T \pm i I = \hspace* {.5mm}/\hspace* {-2.35mm}D+ V \pm i I : \mathcal{K}_1^1(M_0; W \otimes E) \to \mathcal{K}_0^0(M_0; W \otimes E)= L^2(M_0; W \otimes E) $$

are invertible, and hence T is essentially self-adjoint with domain $\mathcal{K}_{1}^{1}(M_{0}; W \otimes E)$, where all the L ² and Sobolev spaces are associated with $\mathcal{W}$.

Proof

Let us denote by

$$ Q_a := f^{1/2}Tf^{1/2} = \hspace* {.5mm}/\hspace* {-2.35mm}D_{\mathcal{W}} + V_0 : \mathcal{K}_{a}^{1/2}(M_0; W \otimes E)\to \mathcal{K}_{a}^{-1/2}(M_0; W \otimes E). $$

Then Q ₀ is fully elliptic (by Lemma 4.5 and the fact that $\hspace* {.5mm}/\hspace* {-2.35mm}D_{\mathcal{W}}$ is the Dirac operator associated with $\mathcal{W}= f\mathcal{V}$), and hence it is Fredholm. It follows from Lemma 4.11 that Q _a is Fredholm for any a and that its index is independent of a. Since $Q_{0}^{*} = Q_{0}$, we have $\operatorname {ind}(Q_{a}) = 0$ for all a. Hence

$$ \operatorname {ind}(Q_a + \lambda f) = \operatorname {ind}(Q_a)=0, $$

since multiplication by f is a compact operator $\mathcal{K}_{a}^{1/2}(M_{0}; W \otimes E) \to \mathcal{K}_{a}^{-1/2}(M_{0}; W \otimes E)$ by [3]. Then

$$ T \pm i I = f^{-1/2}(Q_a \pm i f) f^{-1/2} : \mathcal{K}_{1/2}^{1/2}(M_0; W \otimes E) \to \mathcal{K}_{1/2}^{-1/2}(M_0; W \otimes E) $$

(41)

is also Fredholm of index zero.

We recall that $\mathcal{K}^{m}_{a}(M_{0}; W \otimes E)$ is the dual of $\mathcal{K}^{-m}_{-a}(M_{0}; W \otimes E)$, with the duality pairing being obtained from the L ²-inner product by continuous extension. Then the “L ²-estimate”

$$ ( (T \pm i)u, u) = (Tu, u) \pm i(u, u) $$

and (Tu,u)∈ℝ (since T is symmetric between the indicated spaces in Eq. (41)) show that T±iI are injective for a=0. Since they have index zero, they induce isomorphisms

$$ T \pm i I : \mathcal{K}_{a + 1/2}^{m+1/2}(M_0; W \otimes E) \to \mathcal{K}_{a -1/2}^{m - 1/2}(M_0; W \otimes E) $$

(42)

for a=0 and m=0.

Now for an arbitrary a, the induced operator will still have index zero (by Lemma 4.11). Since for a≥0 it will still be injective, it follows that it will be an isomorphism for all a≥0. Since for a<0 the resulting map is dual to the one for −a, we obtain that T±iI of Eq. (42) are isomorphisms for all a and m=0. We can extend this isomorphism to any m≥0 by elliptic regularity (Lemma 4.10) and this completes the proof by taking a=m=1/2. □

We shall extend T to a self-adjoint operator denoted by the same letter. We are ready now to compute the index of

$$ T^+ = (\hspace* {.5mm}/\hspace* {-2.35mm}D+ V)^+ : \mathcal{K}_1^1(M_0; (W\widehat{\otimes}E)^+) \to \mathcal{K}^0_0(M_0; (W\widehat{\otimes}E)^-), $$

where $\hspace* {.5mm}/\hspace* {-2.35mm}D$ is the Dirac operator on the (arbitrary) even-dimensional Lie manifold $(M_{0}, \mathcal{V})$ and V=f ⁻¹ V ₀ is an unbounded potential as in (38). Let $\pi: \overline{TM} \to M$ and $\pi_{\varOmega}: \varOmega= \partial {\overline{A_{\mathcal{V}}}} \to M$ be the natural projections and Td(T _ℂ M) be the Todd class of the complexified tangent bundle of M.

Theorem 4.13

The operator $T^{+} = (\hspace* {.5mm}/\hspace* {-2.35mm}D+ V)^{+}$ is Fredholm and its index is given by

Proof

Let Q ₁=T ⁺ f, where $f = \prod x_{k}^{a_{k}}$ as above and is regarded as a multiplication operator. Then Q ₁=f ^−1/2 Q ⁺ f ^1/2, where

$$ Q^+ := f^{1/2} T^{+} f^{1/2} = (\hspace* {.5mm}/\hspace* {-2.35mm}D_{\mathcal{W}} + V_0)^+ : \mathcal{K}_{0}^1(M_0; (W\widehat{\otimes}E)^+) \to \mathcal{K}_0^0(M_0; (W\widehat{\otimes}E)^-) $$

is fully elliptic (by Theorem 4.7 and the fact that $\hspace* {.5mm}/\hspace* {-2.35mm}D_{\mathcal{W}}$ is the Dirac operator associated with $\mathcal{W}:= f\mathcal{V}$). Then the operators Q ₁ and Q ⁺ have the same index, by Lemma 4.11. By ellipticity,

$$ (1 + Q_1^*Q_1)^{1/2} : \mathcal{K}_0^1(M_0; (W\widehat{\otimes}E)^+) \to L^2(M_0; (W\widehat{\otimes}E)^+) $$

is an isomorphism since the domain of any elliptic operator $P \in \varPsi^{m}_{\mathcal{W}}(M; E)$ is $H^{m}_{\mathcal{W}}(M_{0}, E)$. Therefore, $f(1\! +\! Q_{1}^{*}Q_{1})^{-1/2} \!:\! L^{2}(M_{0}; (W\widehat{\otimes}E)^{+}) \!\to\!\mathcal{K}_{1}^{1}(M_{0}; (W\widehat{\otimes}E)^{+})$ is an isomorphism as well. Proposition 4.12 then yields that $T^{+} : \mathcal{K}_{1}^{1}(M_{0}) \to L^{2}(M_{0})$ has the same index as

and, in particular, they are both Fredholm.

We have thus obtained that the operator $T^{+} = (\hspace* {.5mm}/\hspace* {-2.35mm}D+ V)^{+}$ is Fredholm and has the same index as Q ⁺:=f ^1/2 T ⁺ f ^1/2. Moreover, the principal symbols of T ⁺ and Q ⁺ define the same K-theory classes, by homotopy invariance, as do the symbols of $\hspace* {.5mm}/\hspace* {-2.35mm}D$ and $\hspace* {.5mm}/\hspace* {-2.35mm}D_{\mathcal{W}}$, and hence

by Theorem 4.7 applied to Q ⁺ and homotopy invariance. □

We also obtain the following more explicit calculation similar to Corollary 4.8.

Corollary 4.14

Let $\hspace* {.5mm}/\hspace* {-2.35mm}D_{F}$ be the Dirac operator twisted with F and $T = \hspace* {.5mm}/\hspace* {-2.35mm}D_{F} + V$ be the perturbed twisted Dirac operator associated with V=f ⁻¹ V ₀, where V ₀ is a bounded potential on M invertible at ∂M for a spin ^c Lie manifold $(M, \mathcal{V})$. Then T ⁺ is Fredholm and, using the notation of Corollary 4.14,

$$ \operatorname {ind}(T^+) = \widehat{A}(M) \operatorname {ch}_0 ([F \otimes V_0] ) [M] = \widehat{A}(M) \operatorname {ch}_0 ([F \otimes V] ) [M]. $$