Abstract
We study \(H=D^*D+V\), where D is a first order elliptic differential operator acting on sections of a Hermitian vector bundle over a Riemannian manifold M, and V is a Hermitian bundle endomorphism. In the case when M is geodesically complete, we establish the essential self-adjointness of positive integer powers of H. In the case when M is not necessarily geodesically complete, we give a sufficient condition for the essential self-adjointness of H, expressed in terms of the behavior of V relative to the Cauchy boundary of M.
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1 Introduction
As a fundamental problem in mathematical physics, self-adjointness of Schrödinger operators has attracted the attention of researchers over many years now, resulting in numerous sufficient conditions for this property in \(L^2(\mathbb {R}^{n})\). For reviews of the corresponding results, see, for instance, the books [14, 28].
The study of the corresponding problem in the context of a non-compact Riemannian manifold was initiated by Gaffney [15, 16] with the proof of the essential self-adjointness of the Laplacian on differential forms. About two decades later, Cordes (see Theorem 3 in [11]) proved the essential self-adjointness of positive integer powers of the operator
on an n-dimensional geodesically complete Riemannian manifold M equipped with a (smooth) metric \(g=(g_{ij})\) [here \((g^{ij})=((g_{ij})^{-1})\)] and a positive smooth measure \(\mathrm{d}\mu \) [i.e. in any local coordinates \(x^{1},\,x^{2},\dots ,x^{n}\) there exists a strictly positive \(C^{\infty }\)-density \(\kappa (x)\) such that \(\mathrm{d}\mu =\kappa (x)\,\mathrm{d}x^{1}\mathrm{d}x^{2}\dots \mathrm{d}x^{n}\)]. Theorem 1 of our paper extends this result to the operator \((D^*D+V)^{k}\) for all \(k\in \mathbb {Z}_{+}\), where D is a first order elliptic differential operator acting on sections of a Hermitian vector bundle over a geodesically complete Riemannian manifold, \(D^{*}\) is the formal adjoint of D, and V is a self-adjoint Hermitian bundle endomorphism; see Sect. 2.2 for details.
In the context of a general Riemannian manifold (not necessarily geodesically complete), Cordes (see Theorem IV.1.1 in [12], Theorem 4 in [11]) proved the essential self-adjointness of \(P^{k}\) for all \(k\in \mathbb {Z}_{+}\), where
and \(q\in C^{\infty }(M)\) is real-valued. Thanks to a Roelcke-type estimate (see Lemma 3.1 below), the technique of Cordes [12] can be applied to the operator \((D^*D+V)^{k}\) acting on sections of Hermitian vector bundles over a general Riemannian manifold. To make our exposition shorter, in Theorem 1 we consider the geodesically complete case. Our Theorem 2 concerns \((\nabla ^*\nabla +V)^{k}\), where \(\nabla \) is a metric connection on a Hermitian vector bundle over a non-compact geodesically complete Riemannian manifold. This result extends Theorem 1.1 of [13] where Cordes showed that if (M, g) is non-compact and geodesically complete and P is semi-bounded from below on \(C_{c}^{\infty }(M)\), then \(P^{k}\) is essentially self-adjoint on \(C_{c}^{\infty }(M)\), for all \(k\in \mathbb {Z}_{+}\).
For the remainder of the introduction, the notation \(D^*D+V\) is used in the same sense as described earlier in this section. In the setting of geodesically complete Riemannian manifolds, the essential self-adjointness of \(D^*D+V\) with \(V\in L^{\infty }_{\mathrm{loc}}\) was established in [20], providing a generalization of the results in [3, 26, 27, 31] concerning Schrödinger operators on functions (or differential forms). Subsequently, the operator \(D^*D+V\) with a singular potential V was considered in [5]. Recently, in the case \(V\in L^{\infty }_{\mathrm{loc}}\), the authors of [4] extended the main result of [5] to the operator \(D^*D+V\) acting on sections of infinite-dimensional bundles whose fibers are modules of finite type over a von Neumann algebra.
In the context of an incomplete Riemannian manifold, the authors of [17, 21, 22] studied the so-called Gaffney Laplacian, a self-adjoint realization of the scalar Laplacian generally different from the closure of \(\Delta _{M,d \mu }|_{C_{c}^{\infty }(M)}\). For a study of Gaffney Laplacian on differential forms, see [23].
Our Theorem 3 gives a condition on the behavior of V relative to the Cauchy boundary of M that will guarantee the essential self-adjointness of \(D^*D+V\); for details see Sect. 2.3 below. Related results can be found in [6, 24, 25] in the context of (magnetic) Schrödinger operators on domains in \({\mathbb {R}}^n\), and in [10] concerning the magnetic Laplacian on domains in \({\mathbb {R}}^n\) and certain types of Riemannian manifolds.
Finally, let us mention that Chernoff [7] used the hyperbolic equation approach to establish the essential self-adjointness of positive integer powers of Laplace–Beltrami operator on differential forms. This approach was also applied in [2, 8, 9, 18, 19, 30] to prove essential self-adjointness of second-order operators (acting on scalar functions or sections of Hermitian vector bundles) on Riemannian manifolds. Additionally, the authors of [18, 19] used path integral techniques.
The paper is organized as follows. The main results are stated in Sect. 2, a preliminary lemma is proven in Sect. 3, and the main results are proven in Sects. 4–6.
2 Main results
2.1 The setting
Let M be an n-dimensional smooth, connected Riemannian manifold without boundary. We denote the Riemannian metric on M by \(g^{\mathrm{TM}}\). We assume that M is equipped with a positive smooth measure \(\mathrm{d}\mu \), i.e. in any local coordinates \(x^{1}, \,x^{2},\dots ,x^{n}\) there exists a strictly positive \(C^{\infty }\)-density \(\kappa (x)\) such that \(\mathrm{d}\mu =\kappa (x)\,\mathrm{d}x^{1}\mathrm{d}x^{2}\dots \mathrm{d}x^{n}\). Let E be a Hermitian vector bundle over M and let \(L^2(E)\) denote the Hilbert space of square integrable sections of E with respect to the inner product
where \(\langle \cdot ,\cdot \rangle _{E_{x}}\) is the fiberwise inner product. The corresponding norm in \(L^2(E)\) is denoted by \(\Vert \cdot \Vert \). In Sobolev space notations \(W^{k,2}_{\mathrm{loc}}(E)\) used in this paper, the superscript \(k\in \mathbb {Z}_{+}\) indicates the order of the highest derivative. The corresponding dual space is denoted by \(W^{-k,2}_{\mathrm{loc}}(E)\).
Let F be another Hermitian vector bundle on M. We consider a first order differential operator \(D:C_{c}^{\infty }(E)\rightarrow C_{c}^{\infty }(F)\), where \(C^\infty _c\) stands for the space of smooth compactly supported sections. In the sequel, by \(\sigma (D)\) we denote the principal symbol of D.
Assumption (A0) Assume that D is elliptic. Additionally, assume that there exists a constant \(\lambda _0>0\) such that
where \(|\xi |\) is the length of \(\xi \) induced by the metric \(g^{\mathrm{TM}}\) and \(|\sigma (D)(x,\xi )|\) is the operator norm of \(\sigma (D)(x,\xi ):E_x\rightarrow F_x\).
Remark 2.1
Assumption (A0) is satisfied if \(D=\nabla \), where \(\nabla :C^\infty (E)\rightarrow C^{\infty }(T^*M\otimes E)\) is a covariant derivative corresponding to a metric connection on a Hermitian vector bundle E over M.
2.2 Schrödinger-type operator
Let \(D^*:C_{c}^{\infty }(F)\rightarrow C_{c}^{\infty }(E)\) be the formal adjoint of D with respect to the inner product (2.1). We consider the operator
where \(V\in L^{\infty }_{\mathrm{loc}}(\mathrm{End}{E})\) is a linear self-adjoint bundle endomorphism. In other words, for all \(x\in M\), the operator \(V(x):E_{x}\rightarrow E_{x}\) is self-adjoint and \(|V(x)|\in L^{\infty }_{\mathrm{loc}}(M)\), where |V(x)| is the norm of the operator \(V(x):E_{x}\rightarrow E_{x}\).
2.3 Statements of results
Theorem 1
Let M, \(g^{\mathrm{TM}}\), and \(\mathrm{d}\mu \) be as in Sect. 2.1. Assume that \((M,g^{\mathrm{TM}})\) is geodesically complete. Let E and F be Hermitian vector bundles over M, and let \(D:C_{c}^{\infty }(E)\rightarrow C_{c}^{\infty }(F)\) be a first order differential operator satisfying the Assumption (A0). Assume that \(V\in C^{\infty }(\mathrm{End}{E})\) and
where C is a constant, and the inequality is understood in operator sense. Then \(H^{k}\) is essentially self-adjoint on \(C_{c}^{\infty }(E)\), for all \(k\in \mathbb {Z}_{+}\).
Remark 2.2
In the case \(V=0\), the following result related to Theorem 1 can be deduced from Chernoff (see Theorem 2.2 in [7]):
Assume that (M, g) is a geodesically complete Riemannian manifold with metric g. Let D be as in Theorem 1, and define
Fix \(x_0\in M\) and define
where \(r>0\) and \(B(x_0,r):=\{x\in M:\mathrm{d}_{g}(x_0,x)<r\}\). Assume that
Then the operator \((D^*D)^{k}\) is essentially self-adjoint on \(C_{c}^{\infty }(E)\) for all \(k\in \mathbb {Z}_{+}\).
At the end of this section we give an example of an operator for which Theorem 1 guarantees the essential self-adjointness of \((D^*D)^{k}\), whereas Chernoff’s result cannot be applied.
The next theorem is concerned with operators whose potential V is not necessarily semi-bounded from below.
Theorem 2
Let M, \(g^{\mathrm{TM}}\), and \(\mathrm{d}\mu \) be as in Sect. 2.1. Assume that \((M,g^{\mathrm{TM}})\) is noncompact and geodesically complete. Let E be a Hermitian vector bundle over M and let \(\nabla \) be a Hermitian connection on E. Assume that \(V\in C^{\infty }(\mathrm{End}{E})\) and
where \(q\in C^{\infty }(M)\) and the inequality is understood in the sense of operators \(E_x\rightarrow E_x\). Additionally, assume that
where \(C\in \mathbb {R}\) and \(\Delta _{M,\mu }\) is as in (1.1) with g replaced by \(g^{\mathrm{TM}}\). Then the operator \((\nabla ^*\nabla +V)^{k}\) is essentially self-adjoint on \(C_{c}^{\infty }(E)\), for all \(k\in \mathbb {Z}_{+}\).
Remark 2.3
Let us stress that non-compactness is required in the proof to ensure the existence of a positive smooth solution of an equation involving \(\Delta _{M,\mu }+q\). In the case of a compact manifold, such a solution exists under an additional assumption; see Theorem III.6.3 in [12].
In our last result we will need the notion of Cauchy boundary. Let \(d_{g^{\mathrm{TM}}}\) be the distance function corresponding to the metric \(g^{\mathrm{TM}}\). Let \((\widehat{M}, \widehat{d}_{g^{\mathrm{TM}}})\) be the metric completion of \((M, d_{g^{\mathrm{TM}}})\). We define the Cauchy boundary \(\partial _{C}M\) as follows: \(\partial _{C}M:=\widehat{M}\backslash M\). Note that \((M,d_{g^{\mathrm{TM}}})\) is metrically complete if and only if \(\partial _{C}M\) is empty. For \(x\in M\) we define
We will also need the following assumption:
Assumption (A1) Assume that \(\widehat{M}\) is a smooth manifold and that the metric \(g^{\mathrm{TM}}\) extends to \(\partial _{C}M\).
Remark 2.4
Let N be a (smooth) n-dimensional Riemannian manifold without boundary. Denote the metric on N by \(g^{TN}\) and assume that \((N,g^{TN})\) is geodesically complete. Let \(\Sigma \) be a k-dimensional closed sub-manifold of N with \(k<n\). Then \(M:=N\backslash \Sigma \) has the properties \(\widehat{M}=N\) and \(\partial _{C}M=\Sigma \). Thus, Assumption (A1) is satisfied.
Theorem 3
Let M, \(g^{\mathrm{TM}}\), and \(\mathrm{d}\mu \) be as in Sect. 2.1. Assume that (A1) is satisfied. Let E and F be Hermitian vector bundles over M, and let \(D:C_{c}^{\infty }(E)\rightarrow C_{c}^{\infty }(F)\) be a first order differential operator satisfying the Assumption (A0). Assume that \(V\in L^{\infty }_{\mathrm{loc}}(\mathrm{End}{E})\) and there exists a constant C such that
where \(\lambda _0\) is as in (2.2), the distance r(x) is as in (2.7), and the inequality is understood in the sense of linear operators \(E_{x}\rightarrow E_{x}\). Then H is essentially self-adjoint on \(C_{c}^{\infty }(E)\).
In order to describe the example mentioned in Remark 2.2, we need the following
Remark 2.5
As explained in [5], we can use a first-order elliptic operator \(D:C_{c}^{\infty }(E)\rightarrow C_{c}^{\infty }(F)\) to define a metric on M. For \(\xi ,\eta \in T^*_xM\), define
where \({\text {Tr}}\) denotes the usual trace of a linear operator. Since D is an elliptic first-order differential operator and \(\sigma (D)(x,\xi )\) is linear in \(\xi \), it is easily checked that (2.9) defines an inner product on \(T^*_xM\). Its dual defines a Riemannian metric on M. Denoting this metric by \(g^{\mathrm{TM}}\) and using elementary linear algebra, it follows that (2.2) is satisfied with \(\lambda _0=\sqrt{m}\).
Example 2.6
Let \(M=\mathbb {R}^{2}\) with the standard metric and measure, and \(V=0\). Denoting respectively by \(C_{c}^{\infty }(\mathbb {R}^{2};\mathbb {R})\) and \(C_{c}^{\infty }(\mathbb {R}^{2};\mathbb {R}^2)\) the spaces of smooth compactly supported functions \(f:\mathbb {R}^{2}\rightarrow \mathbb {R}\) and \(f:\mathbb {R}^{2}\rightarrow \mathbb {R}^2\), we define the operator \(D:C_{c}^{\infty }(\mathbb {R}^{2};\mathbb {R})\rightarrow C_{c}^{\infty }(\mathbb {R}^{2};\mathbb {R}^2)\) by
where
Since a, b are smooth real-valued nowhere vanishing functions in \(\mathbb {R}^2\), it follows that the operator D is elliptic. We are interested in the operator
The matrix of the inner product on \(T^*M\) defined by D via (2.9) is \(\mathrm{diag}(a^2/2,b^2/2)\). The matrix of the corresponding Riemannian metric \(g^{\mathrm{TM}}\) on M is \(\mathrm{diag}(2a^{-2},2b^{-2})\), so the metric itself is \(ds^2=2a^{-2}\mathrm{d}x^2+2b^{-2}dy^2\) and it is geodesically complete (see Example 3.1 of [5]). Moreover, thanks to Remark 2.5, Assumption (A0) is satisfied. Thus, by Theorem 1 the operator \((D^*D)^{k}\) is essentially self-adjoint for all \(k\in \mathbb {Z}_{+}\). Furthermore, in Example 3.1 of [5] it was shown that for the considered operator D the condition (2.4) is not satisfied. Thus, the result stated in Remark 2.2 does not apply.
3 Roelcke-type inequality
Let M, \(\mathrm{d}\mu \), D, and \(\sigma (D)\) be as in Sect. 2.1. Set \(\widehat{D}:=-i\sigma (D)\), where \(i=\sqrt{-1}\). Then for any Lipschitz function \(\psi :M\rightarrow \mathbb {R}\) and \(u\in W^{1,2}_{\mathrm{loc}}(E)\) we have
where we have suppressed x for simplicity. We also note that \(\widehat{D^*}(\xi )=-(\widehat{D}(\xi ))^{*}\), for all \(\xi \in T_{x}^{*}M\).
For a compact set \(K\subset M\), and \(u,\, v\in W^{1,2}_{\mathrm{loc}}(E)\), we define
In order to prove Theorem 1 we need the following important lemma, which is an extension of Lemma IV.2.1 in [12] to operator (2.3). In the context of the scalar Laplacian on a Riemannian manifold, this kind of result is originally due to Roelcke [29].
Lemma 3.1
Let M, \(g^{\mathrm{TM}}\), and \(\mathrm{d}\mu \) be as in Sect. 2.1. Let E and F be Hermitian vector bundles over M, and let \(D:C_{c}^{\infty }(E)\rightarrow C_{c}^{\infty }(F)\) be a first order differential operator satisfying the Assumption (A0). Let \(\rho :M\rightarrow [0,\infty )\) be a function satisfying the following properties:
-
(i)
\(\rho (x)\) is Lipschitz continuous with respect to the distance induced by the metric \(g^{\mathrm{TM}}\);
-
(ii)
\(\rho (x_0)=0\), for some fixed \(x_0\in M\);
-
(iii)
the set \(B_{T}:=\{x\in M:\rho (x)\le T\}\) is compact, for some \(T>0\).
Then the following inequality holds for all \(u\in W^{2,2}_{\mathrm{loc}}(E)\) and \(v\in W^{2,2}_{\mathrm{loc}}(E)\):
where \(B_{t}\) is as in (iii) (with t instead of T), the constant \(\lambda _0\) is as in (2.2), and \(|\mathrm{d}\rho (x)|\) is the length of \(\mathrm{d}\rho (x)\in T_{x}^*M\) induced by \(g^{\mathrm{TM}}\).
Proof
For \(\varepsilon >0\) and \(t\in (0,T)\), we define a continuous piecewise linear function \(F_{\varepsilon ,t}\) as follows:
The function \(f_{\varepsilon ,t}(x):=F_{\varepsilon ,t}(\rho (x))\), is Lipschitz continuous with respect to the distance induced by the metric \(g^{\mathrm{TM}}\), and \(df_{\varepsilon ,t}(x)=(F'_{\varepsilon ,t}(\rho (x)))\mathrm{d}\rho (x)\). Moreover we have \(f_{\varepsilon ,t}v\in W^{1,2}_{\mathrm{loc}}(E)\) for all \(v\in W^{1,2}_{\mathrm{loc}}(E)\), since
It follows from the compactness of \(B_{T}\) that \(B_{t}\) is compact for all \(t\in (0,T)\). Using integration by parts (see Lemma 8.8 in [5]), for all \(u\in W^{2,2}_{\mathrm{loc}}(E)\) and \(v\in W^{2,2}_{\mathrm{loc}}(E)\) we have
which, together with (2.2), gives
where \(|df_{\varepsilon ,t}(x)|\) and \(|\mathrm{d}\rho (x)|\) are the norms of \(df_{\varepsilon ,t}(x)\in T_{x}^*M\) and \(\mathrm{d}\rho (x)\in T_{x}^*M\) induced by \(g^{\mathrm{TM}}\).
Fixing \(\varepsilon >0\), integrating the leftmost and the rightmost side of (3.4) from \(t=0\) to \(t=T\), and noting that \(F'_{\varepsilon ,t}(\rho (x))\) is the only term on the rightmost side depending on t, we obtain
where
We now let \(\varepsilon \rightarrow 0+\) in (3.5). On the left-hand side of (3.5), as \(\varepsilon \rightarrow 0+\), we have \(f_{\varepsilon ,t}(x)\rightarrow \chi _{B_t}(x)\) almost everywhere, where \(\chi _{B_t}(x)\) is the characteristic function of the set \(B_t\). Additionally, \(|f_{\varepsilon ,t}(x)|\le 1\) for all \(x\in B_t\) and all \(t\in (0,T)\); thus, by dominated convergence theorem, as \(\varepsilon \rightarrow 0+\) the left-hand side of (3.5) converges to the left-hand side of (3.3). On the right-hand side of (3.5) an easy calculation shows that \(I_{\varepsilon }(x)\rightarrow 1\), as \(\varepsilon \rightarrow 0+\). Additionally, we have \(|I_{\varepsilon }(x)|\le 1\), a.e. on \(B_{T}\); hence, by the dominated convergence theorem, as \(\varepsilon \rightarrow 0+\) the right-hand side of (3.5) converges to the right-hand side of (3.3). This establishes the inequality (3.3). \(\square \)
4 Proof of Theorem 1
We first give the definitions of minimal and maximal operators associated with the expression H in (2.3).
4.1 Minimal and maximal operators
We define \(H_{\min }u:=Hu\), with \({\text {Dom}}(H_{\min }):=C_{c}^{\infty }(E)\), and \(H_{\max }:=(H_{\min })^{*}\), where \(T^*\) denotes the adjoint of operator T. Denoting \(\mathscr {D}_{\max }:=\{u\in L^2(E):Hu\in L^2(E)\}\), we recall the following well-known property: \({\text {Dom}}(H_{\max })=\mathscr {D}_{\max }\) and \(H_{\max }u=Hu\) for all \(u\in \mathscr {D}_{\max }\).
From now on, throughout this section, we assume that the hypotheses of Theorem 1 are satisfied. Let \(x_0\in M\), and define \(\rho (x):=d_{g^{\mathrm{TM}}}(x_0,x)\), where \(d_{g^{\mathrm{TM}}}\) is the distance function corresponding to the metric \(g^{\mathrm{TM}}\). By the definition of \(\rho (x)\) and the geodesic completeness of \((M, g^{\mathrm{TM}})\), it follows that \(\rho (x)\) satisfies all hypotheses of Lemma 3.1. Using Lemma 3.1 and Proposition 4.1 below, we are able to apply the method of Cordes [11, 12] to our context. As we will see, Cordes’s technique reduces our problem to a system of ordinary differential inequalities of the same type as in Section IV.3 of [12].
Proposition 4.1
Let A be a densely defined operator with domain \(\mathscr {D}\) in a Hilbert space \(\mathscr {H}\). Assume that A is semi-bounded from below, that \(A\mathscr {D}\subseteq \mathscr {D}\), and that there exists \(c_0\in \mathbb {R}\) such that the following two properties hold:
-
(i)
\(((A+c_0I)u,u)_{\mathscr {H}}\ge \Vert u\Vert _{\mathscr {H}}^2\), for all \(u\in \mathscr {D}\), where I denotes the identity operator in \(\mathscr {H}\);
-
(ii)
\((A+c_0I)^{k}\) is essentially self-adjoint on \(\mathscr {D}\), for some \(k\in \mathbb {Z}_{+}\).
Then, \((A+cI)^{j}\) is essentially self-adjoint on \(\mathscr {D}\), for all \(j=1,2,\dots , k\) and all \(c\in \mathbb {R}\).
Remark 4.2
To prove Proposition 4.1, one may mimick the proof of Proposition IV.1.4 in [12], which was carried out for the operator P defined in (1.2) with \(\mathscr {D}=C_{c}^{\infty }(M)\), since only abstract functional analysis facts and the property \(P\mathscr {D}\subseteq \mathscr {D}\) were used.
We start the proof of Theorem 1 by noticing that the operator \(H_{\min }\) is essentially self-adjoint on \(C_{c}^{\infty }(E)\); see Corollary 2.9 in [5]. Thanks to Proposition 4.1, whithout any loss of generality we can change V(x) to \( V(x) + C\,\text {Id}(x)\) , where C is a sufficiently large constant in order to have
where \(\lambda _0\) is as in (2.2) and \(\text {Id}(x)\) is the identity endomorphism of \(E_{x}\). Using non-negativity of \(D^*D\) and (4.1) we have
which leads to
and, hence, \(\Vert Hu\Vert \ge \Vert u\Vert \), for all \(u\in C_{c}^{\infty }(E)\). Therefore,
and
By (4.3) we have
and, hence, \(\Vert H^2u\Vert \ge \Vert u\Vert \), for all \(u\in C_{c}^{\infty }(E)\). This, in turn, leads to
Continuing like this, we obtain \((H^ku,u)\ge \Vert u\Vert ^2\), for all \(u\in C_{c}^{\infty }(E)\) and all \(k\in \mathbb {Z}_{+}\). In this case, by an abstract fact (see Theorem X.26 in [28]), the essential self-adjointness of \(H^{k}\) on \(C_{c}^{\infty }(E)\) is equivalent to the following statement: if \(u\in L^2(E)\) satisfies \(H^{k}u=0\), then \(u=0\).
Let \(u\in L^2(E)\) satisfy \(H^{k}u=0\). Since \(V\in C^{\infty }(E)\), by local elliptic regularity it follows that \(u\in C^{\infty }(E)\cap L^2(E)\). Define
Here, in the case \(k-j<0\), the definition (4.4) is interpreted as \(((H_{\max })^{-1})^{j-k}\). We already noted that \(H_{\min }\) is essentially self-adjoint and positive. Furthermore, it is well known that the self-adjoint closure of \(H_{\min }\) coincides with \(H_{\max }\). Therefore \(H_{\max }\) is a positive self-adjoint operator, and \((H_{\max })^{-1}:L^2(E)\rightarrow L^2(E)\) is bounded. This, together with \(f_k=u\in L^2(E)\) explains the following property: \(f_j\in L^2(E)\), for all \(j\ge k\). Additionally, observe that \(f_j= 0\) for all \(j\le 0\) because \(f_0= 0\). Furthermore, we note that \(f_j\in C^{\infty }(E)\), for all \(j\in \mathbb {Z}\). The last assertion is obvious for \(j\le k\), and for \(j>k\) it can be seen by showing that \(H^{j}f_j=0\) in distributional sense and using \(f_j\in L^2(E)\) together with local elliptic regularity. To see this, let \(v\in C_{c}^{\infty }(E)\) be arbitrary, and note that
Finally, observe that
With \(f_j\) as in (4.4), define the functions \(\alpha _j\) and \(\beta _j\) on the interval \(0\le T<\infty \) by the formulas
where \(\lambda _0\) is as in (4.1) and \((\cdot ,\cdot )_{B_t}\) is as in (3.2).
In the sequel, to simplify the notations, the functions \(\alpha _j(T)\) and \(\beta _j(T)\), the inner products \((\cdot ,\cdot )_{B_{t}}\), and the corresponding norms \(\Vert \cdot \Vert _{B_t}\) appearing in (4.6) will be denoted by \(\alpha _j\), \(\beta _j\), \((\cdot ,\cdot )_{t}\), and \(\Vert \cdot \Vert _{t}\), respectively.
Note that \(\alpha _j\) and \(\beta _j\) are absolutely continuous on \([0,\infty )\). Furthermore, \(\alpha _j\) and \(\beta _j\) have a left first derivative and a right first derivative at each point. Additionally, \(\alpha _j\) and \(\beta _j\) are differentiable, except at (at most) countably many points. In the sequel, to simplify notations, we shall denote the right first derivatives of \(\alpha _j\) and \(\beta _j\) by \(\alpha _j'\) and \(\beta _j'\). Note that \(\alpha _j\), \(\beta _j\), \(\alpha _j'\) and \(\beta _j'\) are non-decreasing and non-negative functions. Note also that \(\alpha _j\) and \(\beta _j\) are convex functions. Furthermore, since \(f_j=0\) for all \(j\le 0\), it follows that \(\alpha _j\equiv 0\) and \(\beta _j\equiv 0\) for all \(j\le 0\). Finally, using (4.1) and the property \(f_j\in L^2(E)\cap C^{\infty }(E)\) for all \(j\ge k\), observe that
for all \(j>k\). Here, “integration by parts” in the first equality is justified because \(H_{\min }\) is essentially self-adjoint (i.e. \(C_{c}^{\infty }(E)\) is an operator core of \(H_{\max }\)). Hence, \(\alpha _j'\) and \(\beta _j'\) are bounded for all \(j>k\). It turns out that \(\alpha _j\) and \(\beta _j\) satisfy a system of differential inequalities, as seen in the next proposition.
Proposition 4.3
Let \(\alpha _j\) and \(\beta _j\) be as in (4.6). Then, for all \(j\ge 1\) and all \(T\ge 0\) we have
and
where \(\lambda _0\) is as in (4.1) and \(\alpha _i'\), \(\beta _i'\) denote the right-hand derivatives.
Remark 4.4
Note that the sums in (4.7) and (4.8) are finite since \(\alpha _{i}\equiv 0\) and \(\beta _{i}\equiv 0\) for \(i\le 0\). As our goal is to show that \(f_k=u=0\), we will only use the first k inequalities in (4.7) and the first k inequalities in (4.8).
Proof of Proposition 4.3
From (4.6) and (4.1) it follows that
We start from (4.9), use (3.3), Cauchy–Schwarz inequality, and (4.5) to obtain
We continue the process as follows:
where we used triangle inequality, (3.3), Cauchy–Schwarz inequality, and (4.5). We continue like this until the last term reaches the subscript \(j-l\le 0\), which makes the last term equal zero by properties of \(f_{i}\) discussed above. This establishes (4.7).
To show (4.8), we start from the definition of \(\alpha _j\), use (3.3), Cauchy–Schwarz inequality, and (4.5) to obtain
We continue like this until the last term reaches the subscript \(j-l\le 0\), which makes the last term equal zero by properties of \(f_{i}\) discussed above. This establishes (4.8). \(\square \)
End of the proof of Theorem 1
We will now transform the system (4.7) and (4.8) by introducing new variables:
To carry out the transformation, observe that Cauchy–Schwarz inequality applied to vectors \(\Big \langle \sqrt{\alpha '_i}, \sqrt{\beta '_i}\Big \rangle \) and \(\Big \langle \sqrt{\beta '_p},\sqrt{\alpha '_p}\Big \rangle \) in \(\mathbb {R}^2\) gives
which, together with (4.7) and (4.8) leads to
and
where \(\lambda _0\) is as in (4.1) and \(\omega _i'\), \(\theta _i'\) denote the right-hand derivatives.
The functions \(\omega _j\) and \(\theta _j\) satisfy the following properties: (i) \(\omega _j\) and \(\theta _j\) are absolutely continuous on \([0,\infty )\), and the right-hand derivatives \(\omega _j'\) and \(\theta _j'\) exist everywhere; (ii) \(\omega _j\) and \(\omega _j'\) are non-negative and non-increasing; (iii) \(\omega _j\) is convex; (iv) \(\omega _{j}'\) is bounded for all \(j\ge k\); (v) \(\omega _j(0)=\theta _j(0)=0\); and (vi) \(|\theta _j(T)|\le \omega _j(T)\) and \(|\theta _j'(T)|\le \omega _j'(T)\) for all \(T\in [0,\infty )\).
In Section IV.3 of [12] it was shown that if \(\omega _j\) and \(\theta _j\) are functions satisfying the above described properties (i)–(vi) and the system (4.11) and (4.12), then \(\omega _{j}\equiv 0\) for all \(j=1,2,\dots , k\). In particular, we have \(\omega _k(T)=0\), for all \(T\in [0,\infty )\), and hence \(f_k=0\). Going back to (4.4), we get \(u=0\), and this concludes the proof of essential self-adjointness of \(H^{k}\) on \(C_{c}^{\infty }(E)\). The essential self-adjointness of \(H^2\), \(H^{3}\), \(\dots \), and \(H^{k-1}\) on \(C_{c}^{\infty }(E)\) follows by Proposition 4.1. \(\square \)
5 Proof of Theorem 2
We adapt the proof of Theorem 1.1 in [13] to our type of operator. By assumption (2.6) it follows that
Since (5.1) is satisfied and since M is non-compact and \(g^{\mathrm{TM}}\) is geodesically complete, a result of Agmon [1] (see also Proposition III.6.2 in [12]) guarantees the existence of a function \(\gamma \in C^{\infty }(M)\) such that \(\gamma (x)>0\) for all \(x\in M\), and
We now use the function \(\gamma \) to transform the operator \(H=\nabla ^*\nabla +V\). Let \(L_{\mu _1}^2(E)\) be the space of square integrable sections of E with inner product \((\cdot ,\cdot )_{\mu _1}\) as in (2.1), where \(\mathrm{d}\mu \) is replaced by \(\mathrm{d}\mu _1:=\gamma ^2 \mathrm{d}\mu \). For clarity, we denote \(L^2(E)\) from Sect. 2.1 by \(L_{\mu }^2(E)\). In what follows, the formal adjoints of \(\nabla \) with respect to inner products \((\cdot ,\cdot )_{\mu }\) and \((\cdot ,\cdot )_{\mu _1}\) will be denoted by \(\nabla ^{*,\mu }\) and \(\nabla ^{*,\mu _1}\), respectively. It is easy to check that the map \(T_{\gamma }:L_{\mu }^2(E)\rightarrow L_{\mu _1}^2(E)\) defined by \(Tu:=\gamma ^{-1} u\) is unitary. Furthermore, under the change of variables \(u\mapsto \gamma ^{-1} u\), the differential expression \(H=\nabla ^{*,\mu }\nabla +V\) gets transformed into \(H_1:=\gamma ^{-1}H\gamma \). Since T is unitary, the essential self-adjointness of \(H^k|_{C_{c}^{\infty }(E)}\) in \(L_{\mu }^2(E)\) is equivalent to essential self-adjointness of \((H_{1})^k|_{C_{c}^{\infty }(E)}\) in \(L_{\mu _1}^2(E)\).
In the sequel, we will show that \(H_1\) has the following form:
with
To see this, let \(w,\,z\in C_{c}^{\infty }(E)\) and consider
Setting \(\xi :=d(\gamma ^2/2)\in T^*M\) and using equation (1.34) in Appendix C of [32] we have
where X is the vector field associated with \(\xi \in T^*M\) via the metric \(g^{\mathrm{TM}}\).
Furthermore, by equation (1.35) in Appendix C of [32] we have
where, in local coordinates \(x^{1},\,x^{2},\dots ,x^{n}\), for \(X=X^j\frac{\partial }{\partial x^{j}}\), with Einstein summation convention,
[Recall that \(\mathrm{d}\mu =\kappa (x)\,\mathrm{d}x^{1}\mathrm{d}x^{2}\dots \mathrm{d}x^{n}\), where \(\kappa (x)\) is a positive \(C^{\infty }\)-density.] Since \(X^{j}=(g^{\mathrm{TM}})^{jl}\left( \gamma \frac{\partial \gamma }{\partial x^{l}}\right) \), we have
where \(|\mathrm{d}\gamma (x)|\) is the norm of \(\mathrm{d}\gamma (x)\in T_{x}^*M\) induced by \(g^{\mathrm{TM}}\), and \(\Delta _{M,\mu }\) is as in (1.1) with metric \(g^{\mathrm{TM}}\). Combining (5.4)–(5.7) and noting that
we obtain
which shows (5.3).
By (2.5) and (5.2) it follows that
where C is as in (2.6). Thus, by Theorem 1 the operator \((H_1)^{k}|_{C_{c}^{\infty }(E)}\) is essentially self-adjoint in \(L^2_{\mu _1}(E)\) for all \(k\in \mathbb {Z}_{+}\). \(\square \)
6 Proof of Theorem 3
Throughout the section, we assume that the hypotheses of Theorem 3 are satisfied. In subsequent discussion, the notation \(\widehat{D}\) is as in (3.1) and the operators \(H_{\min }\) and \(H_{\max }\) are as in Sect. 4.1. We begin with the following lemma, whose proof is a direct consequence of the definition of \(H_{\max }\) and local elliptic regularity.
Lemma 6.1
Under the assumption \(V\in L^{\infty }_{\mathrm{loc}}(\mathrm{End}E)\), we have the following inclusion: \({\text {Dom}}(H_{\max })\subset W^{2,2}_{\mathrm{loc}}(E)\).
The proof of the next lemma is given in Lemma 8.10 of [5].
Lemma 6.2
For any \(u\in {\text {Dom}}(H_{\max })\) and any Lipschitz function with compact support \(\psi :M\rightarrow \mathbb {R}\), we have:
Corollary 6.3
Let H be as in (2.3), let \(u\in L^2(E)\) be a weak solution of \(Hu=0\), and let \(\psi :M\rightarrow \mathbb {R}\) be a Lipschitz function with compact support. Then
where \((\cdot ,\cdot )\) on the left-hand side denotes the duality between \(W_{\mathrm{loc}}^{1,2}(E)\) and \(W^{-1,2}_{\mathrm{comp}}(E)\).
Proof
Since \(u\in L^2(E)\) and \(Hu=0\), we have \(u\in {\text {Dom}}(H_{\max })\subset W^{2,2}_{\mathrm{loc}}(E)\subset W^{1,2}_{\mathrm{loc}}(E)\), where the first inclusion follows by Lemma 6.1. Since \(\psi \) is a Lipschitz compactly supported function, we get \(\psi u\in W^{1,2}_{\mathrm{comp}}(E)\) and, hence, \(H(\psi u)\in W^{-1,2}_{\mathrm{comp}}(E)\). Now the equality (6.2) follows from (6.1), the assumption \(Hu=0\), and
where in the second equality we used integration by parts; see Lemma 8.8 in [5]. Here, the two leftmost symbols \((\cdot ,\cdot )\) denote the duality between \(W_{\mathrm{comp}}^{1,2}(E)\) and \(W^{-1,2}_{\mathrm{loc}}(E)\), while the remaining ones stand for \(L^2\)-inner products. \(\square \)
The key ingredient in the proof of Theorem 3 is the Agmon-type estimate given in the next lemma, whose proof, inspired by an idea of [24], is based on the technique developed in [10] for magnetic Laplacians on an open set with compact boundary in \(\mathbb {R}^{n}\).
Lemma 6.4
Let \(\lambda \in \mathbb {R}\) and let \(v\in L^2(E)\) be a weak solution of \((H-\lambda )v=0\). Assume that that there exists a constant \(c_1>0\) such that, for all \(u \in W_{\mathrm{comp}}^{1,2}(E)\),
where r(x) is as in (2.7), \(\lambda _0\) is as in (2.2), the symbol \((\cdot ,\cdot )\) on the left-hand side denotes the duality between \(W_{\mathrm{comp}}^{1,2}(E)\) and \(W^{-1,2}_{\mathrm{loc}}(E)\), and \(|\cdot |\) is the norm in the fiber \(E_{x}\).
Then, the following equality holds: \(v=0\).
Proof
Let \(\rho \) and R be numbers satisfying \(0< \rho < 1/2\) and \( 1 < R < +\infty \). For any \(\varepsilon >0\), we define the function \(f_{\varepsilon }:M \rightarrow \mathbb {R}\) by \(f_{\varepsilon }(x)=F_{\varepsilon }(r(x))\), where r(x) is as in (2.7) and \(F_{\varepsilon }:[0,\infty )\rightarrow \mathbb {R}\) is the continuous piecewise affine function defined by
Let us fix \(x_0\in M\). For any \(\alpha >0\), we define the function \(p_{\alpha }:M \rightarrow \mathbb {R}\) by
where \(P_{\alpha }:[0,\infty ) \rightarrow \mathbb {R}\) is the continuous piecewise affine function defined by
Since \(\widehat{d}_{g^{\mathrm{TM}}}(x_0,x)\le d_{g^{\mathrm{TM}}}(x_0,x)\), it follows that the support of \(f_{\varepsilon }p_{\alpha }\) is contained in the set \(B_{\alpha }:=\{x\in M:\widehat{d}_{g^{\mathrm{TM}}}(x_0,x)\le 2/\alpha \}\). By Assumption (A1) we know that \(\widehat{M}\) is a geodesically complete Riemannian manifold. Hence, by Hopf–Rinow Theorem the set \(B_{\alpha }\) is compact. Therefore, the support of \(f_{\varepsilon }p_{\alpha }\) is compact. Additionally, note that \(f_{\varepsilon }p_{\alpha }\) is a \(\beta \)-Lipschitz function (with respect to the distance corresponding to the metric \(g^{\mathrm{TM}}\)) with \(\beta =\frac{{\rho }}{{\rho -\varepsilon }}+\alpha \).
Since \(v\in L^2(E)\) and \((H-\lambda )v=0\), we have \(v\in {\text {Dom}}(H_{\max })\subset W^{2,2}_{\mathrm{loc}}(E)\subset W^{1,2}_{\mathrm{loc}}(E)\), where the first inclusion follows by Lemma 6.1. Since \(f_{\varepsilon }p_{\alpha }\) is a Lipschitz compactly supported function, we get \(f_{\varepsilon }p_{\alpha } v\in W^{1,2}_{\mathrm{comp}}(E)\) and, hence, \(((H-\lambda )(f_{\varepsilon }p_{\alpha } v))\in W^{-1,2}_{\mathrm{comp}}(E)\).
Using (2.2) we have
where \(|\mathrm{d}(f_{\varepsilon }p_{\alpha })(x)|\) is the norm of \(\mathrm{d}(f_{\varepsilon }p_{\alpha })(x)\in T_{x}^*M\) induced by \(g^{\mathrm{TM}}\).
By Corollary 6.3 with \(H-\lambda \) in place of H and the inequality (6.4), we get
On the other hand, using the definitions of \(f_{\varepsilon }\) and \(p_{\alpha }\) and the assumption (6.3) we have
where
In (6.6) and (6.5), the symbol \((\cdot ,\cdot )\) stands for the duality between \(W_{\mathrm{comp}}^{1,2}(E)\) and \(W^{-1,2}_{\mathrm{loc}}(E)\). We now combine (6.6) and (6.5) to get
We fix \(\rho \), R, and \(\varepsilon \), and let \(\alpha \rightarrow 0+\). After that we let \(\varepsilon \rightarrow 0+\). The last step is to do \(\rho \rightarrow 0+ \) and \(R \rightarrow +\infty \). As a result, we get \(v=0\). \(\square \)
End of the proof of Theorem 3
Using integration by parts (see Lemma 8.8 in [5]), we have
where the two leftmost symbols \((\cdot ,\cdot )\) denote the duality between \(W_{\mathrm{comp}}^{1,2}(E)\) and \(W^{-1,2}_{\mathrm{loc}}(E)\), while the remaining ones stand for \(L^2\)-inner products. Hence, by assumption (2.8) we get:
Choosing, for instance, \(\lambda =-C-2\) in (6.7) we get the inequality (6.3) with \(c_1=1\).
Thus, \(H_{\min }-\lambda \) with \(\lambda =-C-2\) is a symmetric operator satisfying \((u,\, (H_{\min }-\lambda ) u)\ge \Vert u\Vert ^2\), for all \(u\in C_{c}^{\infty }(E)\). In this case, it is known (see Theorem X.26 in [28]) that the essential self-adjointness of \(H_{\min }-\lambda \) is equivalent to the following statement: if \(v\in L^2(E)\) satisfies \((H-\lambda )v=0\), then \(v=0\). Thus, by Lemma 6.4, the operator \((H_{\min }-\lambda )\) is essentially self-adjoint. Hence, \(H_{\min }\) is essentially self-adjoint. \(\square \)
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Milatovic, O., Truc, F. Self-adjoint extensions of differential operators on Riemannian manifolds. Ann Glob Anal Geom 49, 87–103 (2016). https://doi.org/10.1007/s10455-015-9482-0
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DOI: https://doi.org/10.1007/s10455-015-9482-0
Keywords
- Essential self-adjointness
- Hermitian vector bundle
- Higher-order differential operator
- Riemannian manifold