Self-adjoint extensions of differential operators on Riemannian manifolds

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.

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

$$\begin{aligned} \Delta _{M,\mu }:= \ - \ \frac{1}{\kappa }\left( \frac{\partial }{\partial x^{i}}\left( \kappa g^{ij}\frac{\partial }{\partial x^{j}}\right) \right) \end{aligned}$$

(1.1)

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

$$\begin{aligned} Pu:=\Delta _{M,\mu }u+qu, \quad u\in C^{\infty }(M), \end{aligned}$$

(1.2)

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

$$\begin{aligned} (u,v) \ = \ \int _{M}\, \langle u(x),v(x)\rangle _{E_{x}}\, \mathrm{d}\mu (x), \end{aligned}$$

(2.1)

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

$$\begin{aligned} |\sigma (D)(x,\xi )| \ \le \ \lambda _0|\xi |,\quad \text {for all }x\in M, \, \xi \in T_{x}^{*}M, \end{aligned}$$

(2.2)

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

$$\begin{aligned} H \ = \ D^*D \ + \ V, \end{aligned}$$

(2.3)

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

$$\begin{aligned} V(x)\ge C,\quad \text {for all }x\in M, \end{aligned}$$

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

$$\begin{aligned} c(x) \ := \ \sup \{|\sigma (D)(x,\xi )|:\, |\xi |_{T_{x}^{*}M}=1\}. \end{aligned}$$

Fix $x_0\in M$ and define

$$\begin{aligned} \widetilde{c}(r):=\sup _{x\in B(x_0,r)}c(x), \end{aligned}$$

where $r>0$ and $B(x_0,r):=\{x\in M:\mathrm{d}_{g}(x_0,x)<r\}$. Assume that

$$\begin{aligned} \int _{0}^{\infty }\frac{1}{\widetilde{c}(r)}\,\mathrm{d}r=\infty . \end{aligned}$$

(2.4)

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

$$\begin{aligned} V(x)\ge q(x),\quad \text {for all }x\in M, \end{aligned}$$

(2.5)

where $q\in C^{\infty }(M)$ and the inequality is understood in the sense of operators $E_x\rightarrow E_x$. Additionally, assume that

$$\begin{aligned} ((\Delta _{M,\mu }+q)u,u)\ge C\Vert u\Vert ^2,\quad \text {for all }u\in C_{c}^{\infty }(M), \end{aligned}$$

(2.6)

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

$$\begin{aligned} r(x):=\inf _{z\in \partial _{C}M}\widehat{d}_{g^{\mathrm{TM}}}(x,z). \end{aligned}$$

(2.7)

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

$$\begin{aligned} V(x) \ge \left( \frac{\lambda _0}{r(x)}\right) ^2-C,\quad \text { for all }x\in M, \end{aligned}$$

(2.8)

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

$$\begin{aligned} \langle \xi ,\eta \rangle \ = \ \frac{1}{m}\, {\text {Re}}\, {\text {Tr}}\left( \left( \sigma (D)(x,\xi )\right) ^*\sigma (D)(x,\eta )\right) , \quad m= \dim E_x, \end{aligned}$$

(2.9)

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

$$\begin{aligned} D \ = \ \left( \begin{array}{c}a(x,y)\frac{\partial }{\partial x}\\ b(x,y)\frac{\partial }{\partial y}\\ \end{array}\right) , \end{aligned}$$

where

$$\begin{aligned} a(x,y)&= (1-\cos (2\pi e^{x}))x^{2}+1; \\ b(x,y)&= (1-\sin (2\pi e^{y}))y^{2}+1. \end{aligned}$$

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

$$\begin{aligned} H \ := \ D^*D \ = \ -\frac{\partial }{\partial x} \left( a^{2}\frac{\partial }{\partial x}\right) - \frac{\partial }{\partial y} \left( b^{2}\frac{\partial }{\partial y}\right) . \end{aligned}$$

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

$$\begin{aligned} D(\psi u) \ = \ \widehat{D}(d\psi )u+\psi Du, \end{aligned}$$

(3.1)

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

$$\begin{aligned} (u,v)_{K}:=\int _{K}\langle u(x),v(x)\rangle \,\mathrm{d}\mu (x), \quad (Du,Dv)_{K}:=\int _{K}\langle Du(x),Dv(x)\rangle \,\mathrm{d}\mu (x). \end{aligned}$$

(3.2)

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 $v\in W^{2,2}_{\mathrm{loc}}(E)$:

$$\begin{aligned} \int _{0}^{T} |(Du,Dv)_{B_t}-(D^*Du,v)_{B_t}|\,\mathrm{d}t\le \lambda _0\int _{B_{T}}|\mathrm{d}\rho (x)||Du(x)||v(x)|\,\mathrm{d}\mu (x), \end{aligned}$$

(3.3)

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:

$$\begin{aligned} F_{\varepsilon ,t}(s)= \left\{ \begin{array}{ll} 1 &{}\quad \mathrm{for~} s<t-\varepsilon \\ (t-s)/{\varepsilon }&{} \quad \mathrm{for~} t-\varepsilon \le s <t \\ 0 &{}\quad \mathrm{for~ } s \ge t. \end{array} \right. \end{aligned}$$

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

$$\begin{aligned} D(f_{\varepsilon ,t}v)=\widehat{D}(df_{\varepsilon ,t})v+f_{\varepsilon ,t}Dv. \end{aligned}$$

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

$$\begin{aligned} (D^*Du, vf_{\varepsilon ,t})_{B_{t}} = (Du, D(vf_{\varepsilon ,t}))_{B_{t}} = (Du,f_{\varepsilon ,t}Dv)_{B_{t}} + (Du,\widehat{D}(df_{\varepsilon ,t})v)_{B_{t}}, \end{aligned}$$

which, together with (2.2), gives

$$\begin{aligned}&|(Du,f_{\varepsilon ,t}Dv)_{B_{t}}-(D^*Du, vf_{\varepsilon ,t})_{B_{t}}|=|(Du,\widehat{D}(df_{\varepsilon ,t})v)_{B_{t}}|\nonumber \\&\quad \le \int _{B_{t}}|Du(x)||\widehat{D}(df_{\varepsilon ,t}(x))v(x)|\,\mathrm{d}\mu (x) \le \lambda _0 \int _{B_{t}}|Du(x)||df_{\varepsilon ,t}(x)||v(x)|\,\mathrm{d}\mu (x)\nonumber \\&\quad = \lambda _0\int _{B_{t}}|Du(x)||F'_{\varepsilon ,t}(\rho (x))||\mathrm{d}\rho (x)||v(x)|\,\mathrm{d}\mu (x)\nonumber \\&\quad \le \lambda _0\int _{B_{T}}|Du(x)||F'_{\varepsilon ,t}(\rho (x))||\mathrm{d}\rho (x)||v(x)|\,\mathrm{d}\mu (x), \end{aligned}$$

(3.4)

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

$$\begin{aligned}&\int _{0}^{T}|(Du,f_{\varepsilon ,t}Dv)_{B_{t}}-(D^*Du, vf_{\varepsilon ,t})_{B_{t}}|\,\mathrm{d}t\nonumber \\&\quad \le \lambda _0\int _{B_{T}}|Du(x)||\mathrm{d}\rho (x)||v(x)|I_{\varepsilon }(x)\,\mathrm{d}\mu (x), \end{aligned}$$

(3.5)

where

$$\begin{aligned} I_{\varepsilon }(x):=\int _{0}^{T}|F'_{\varepsilon ,t}(\rho (x))|\,\mathrm{d}t. \end{aligned}$$

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

$$\begin{aligned} V(x)\ge \,(\lambda _0^2+1)\text {Id}(x),\quad \text {for all }x\in M, \end{aligned}$$

(4.1)

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

$$\begin{aligned} (H_{\min }u,u)\ge \Vert u\Vert ^2,\quad \text {for all }u\in C_{c}^{\infty }(E), \end{aligned}$$

(4.2)

which leads to

$$\begin{aligned} \Vert u\Vert ^2\le (Hu,u)\le \Vert Hu\Vert \Vert u\Vert ,\quad \text {for all }u\in C_{c}^{\infty }(E), \end{aligned}$$

and, hence, $\Vert Hu\Vert \ge \Vert u\Vert$, for all $u\in C_{c}^{\infty }(E)$. Therefore,

$$\begin{aligned} (H^2u,u)=(Hu,Hu)=\Vert Hu\Vert ^2\ge \Vert u\Vert ^2,\quad \text {for all }u\in C_{c}^{\infty }(E), \end{aligned}$$

(4.3)

and

$$\begin{aligned} (H^3u,u)=(HHu,Hu)\ge \Vert Hu\Vert ^2\ge \Vert u\Vert ^2,\quad \text {for all }u\in C_{c}^{\infty }(E). \end{aligned}$$

By (4.3) we have

$$\begin{aligned} \Vert u\Vert ^2\le (H^2u,u)\le \Vert H^2u\Vert \Vert u\Vert , \quad \text {for all }u\in C_{c}^{\infty }(E), \end{aligned}$$

and, hence, $\Vert H^2u\Vert \ge \Vert u\Vert$, for all $u\in C_{c}^{\infty }(E)$. This, in turn, leads to

$$\begin{aligned} (H^4u,u)=(H^2u,H^2u)=\Vert H^2u\Vert ^2\ge \Vert u\Vert ^2,\quad \text {for all }u\in C_{c}^{\infty }(E). \end{aligned}$$

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

$$\begin{aligned} f_{j}:=H^{k-j}u,\quad j=0,\pm 1, \pm 2, \dots \end{aligned}$$

(4.4)

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

$$\begin{aligned} (f_j,H^{j}v)=(H^{k-j}u,H^{j}v)=(u,H^{k}v)=(H^ku,v)=0. \end{aligned}$$

Finally, observe that

$$\begin{aligned} H^{l}f_j=f_{j-l},\quad \text {for all }j\in \mathbb {Z}\text { and } l\in \mathbb {Z}_{+}\cup \{0\}. \end{aligned}$$

(4.5)

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

$$\begin{aligned} \alpha _j(T):=\lambda _0^2\int _{0}^{T}(f_j,f_j)_{B_{t}}\,\mathrm{d}t,\quad \beta _j(T):=\int _{0}^{T}(D f_j,D f_j)_{B_{t}}\,\mathrm{d}t, \end{aligned}$$

(4.6)

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

$$\begin{aligned} \lambda _0^2(f_j,f_j)+(Df_j,D f_j)\le (Vf_j,f_j)+(D f_j,D f_j)=(f_j,Hf_j)=(f_j,f_{j-1})<\infty , \end{aligned}$$

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

$$\begin{aligned} \alpha _j+\beta _j\le \sqrt{\alpha _j'\beta _j'}+\sum _{l=0}^{\infty }\left( \sqrt{\alpha '_{j+l+1}\beta '_{j-l-1}}+\sqrt{\alpha '_{j-l-1}\beta '_{j+l+1}}\right) \end{aligned}$$

(4.7)

and

$$\begin{aligned} \alpha _j\le \lambda _0^2\left( \sum _{l=0}^{\infty }\left( \sqrt{\alpha '_{j+l+1}\beta '_{j-l}}+\sqrt{\alpha '_{j-l}\beta '_{j+l+1}}\right) \right) , \end{aligned}$$

(4.8)

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

$$\begin{aligned} \alpha _j+\beta _j\le \int _{0}^{T}\left( (f_j, Vf_j)_{t}+(D f_j, D f_j)_{t}\right) \,\mathrm{d}t. \end{aligned}$$

(4.9)

We start from (4.9), use (3.3), Cauchy–Schwarz inequality, and (4.5) to obtain

$$\begin{aligned} \alpha _j+\beta _j&\le \int _{0}^{T}((f_j, Vf_j)_{t}+(D f_j, D f_j)_{t})\,\mathrm{d}t\nonumber \\&=\int _{0}^{T}|(f_j, Hf_j)_{t}-(f_j, D^*D f_j)_{t}+(D f_j, D f_j)_{t}|\,\mathrm{d}t\nonumber \\&\le \lambda _0\int _{B_{T}}|D f_j(x)||f_j(x)|\,\mathrm{d}\mu (x)+\int _{0}^{T}|(f_j, Hf_j)_t|\,\mathrm{d}t\nonumber \\&\le \sqrt{\alpha _j'\beta _j'}+ \int _{0}^{T}|(Hf_{j+1}, f_{j-1})_t|\,\mathrm{d}t. \end{aligned}$$

We continue the process as follows:

$$\begin{aligned} \alpha _j+\beta _j&\le \sqrt{\alpha _j'\beta _j'}+\int _{0}^{T}|(Hf_{j+1}, f_{j-1})_t|\,\mathrm{d}t\nonumber \\&=\sqrt{\alpha _j'\beta _j'}+\int _{0}^{T}|(D^*D f_{j+1}, f_{j-1})_t+(f_{j+1}, Vf_{j-1})_{t}|\,\mathrm{d}t\nonumber \\&\le \sqrt{\alpha _j'\beta _j'}+\int _{0}^{T}|(D^*D f_{j+1}, f_{j-1})_t-(D f_{j+1}, D f_{j-1})_{t}|\,\mathrm{d}t\nonumber \\&\quad +\int _{0}^{T}|(D f_{j+1}, D f_{j-1})_{t}-(f_{j+1},D^*D f_{j-1})_{t}|\,\mathrm{d}t +\int _{0}^{T}|(f_{j+1}, Hf_{j-1})_{t}|\,\mathrm{d}t\nonumber \\&\le \sqrt{\alpha _j'\beta _j'} +\sqrt{\alpha _{j+1}'\beta _{j-1}'} +\sqrt{\alpha _{j-1}'\beta _{j+1}'}+\int _{0}^{T}|(Hf_{j+2}, f_{j-2})_{t}|\,\mathrm{d}t, \end{aligned}$$

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

$$\begin{aligned} \alpha _j&= \lambda _0^2\int _{0}^{T}(f_j,f_j)_{t}\,\mathrm{d}t=\lambda _0^2\int _{0}^{T}|(f_j,Hf_{j+1})_{t}|\,\mathrm{d}t\nonumber \\&=\lambda _0^2\int _{0}^{T}|(f_j,D^*D f_{j+1})_{t}+(Vf_j,f_{j+1})_{t}|\,\mathrm{d}t\nonumber \\&\le \lambda _0^2\int _{0}^{T}|(f_j,D^*D f_{j+1})_{t}-(D f_j,D f_{j+1})_{t}|\,\mathrm{d}t\nonumber \\&\quad +\lambda _0^2\int _{0}^{T}|(D f_j,D f_{j+1})_{t}-(D^*D f_j,f_{j+1})_{t}|\,\mathrm{d}t+\lambda _0^2\int _{0}^{T}|(Hf_j,f_{j+1})_{t}|\,\mathrm{d}t\nonumber \\&\le \lambda _0^2\left( \sqrt{\alpha _{j+1}'\beta _{j}'} +\sqrt{\alpha _{j}'\beta _{j+1}'}\right) +\lambda _0^2\int _{0}^{T}|(f_{j-1},f_{j+1})_{t}|\,\mathrm{d}t. \end{aligned}$$

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:

$$\begin{aligned} \omega _j(T):=\alpha _j(T)+\beta _j(T),\quad \theta _j(T):=\alpha _j(T)-\beta _j(T)\quad T\in [0,\infty ). \end{aligned}$$

(4.10)

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

$$\begin{aligned} \sqrt{\alpha _i'\beta _p'}+\sqrt{\alpha _p'\beta _i'}\le \sqrt{\omega _i'\omega _p'}, \end{aligned}$$

which, together with (4.7) and (4.8) leads to

$$\begin{aligned} \omega _j\le \frac{1}{2}\sqrt{(\omega _j')^2-(\theta _j')^2}+\sum _{l=0}^{\infty }\sqrt{\omega '_{j+l+1}\omega '_{j-l-1}} \end{aligned}$$

(4.11)

and

$$\begin{aligned} \frac{1}{2}(\omega _j+\theta _j)\le \lambda _0^2\left( \sum _{l=0}^{\infty }\sqrt{\omega '_{j+l+1}\omega '_{j-l}}\right) , \end{aligned}$$

(4.12)

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

$$\begin{aligned} ((\Delta _{M,\mu }+q-C+1)u,u)\ge \Vert u\Vert ^2,\quad \text {for all }u\in C_{c}^{\infty }(M). \end{aligned}$$

(5.1)

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

$$\begin{aligned} (\Delta _{M,\mu }+q-C+1)\gamma =\gamma . \end{aligned}$$

(5.2)

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:

$$\begin{aligned} H_1=\nabla ^{*,\mu _1}\nabla +\widetilde{V}, \end{aligned}$$

(5.3)

with

$$\begin{aligned} \widetilde{V}(x):=\frac{\Delta _{M,\mu }\gamma }{\gamma }\,\text {Id}(x)+V(x). \end{aligned}$$

To see this, let $w,\,z\in C_{c}^{\infty }(E)$ and consider

$$\begin{aligned}&(H_1w,z)_{\mu _1}=\int _{M}\langle \gamma ^{-1}H(\gamma w) ,z\rangle \,\gamma ^2\mathrm{d}\mu = \int _{M}\langle H(\gamma w) ,\gamma z\rangle \,\mathrm{d}\mu =(H(\gamma w) ,\gamma z)_{\mu }\nonumber \\&=(\nabla (\gamma w),\nabla (\gamma z))_{\mu }+(V\gamma w, \gamma z)_{\mu }=(\gamma ^2\nabla w,\nabla z)_{\mu }+(\mathrm{d}\gamma \otimes w,\mathrm{d}\gamma \otimes z)_{L^2_{\mu }(T^*M\otimes E)}\nonumber \\&\quad +(\gamma \nabla w,\mathrm{d}\gamma \otimes z)_{L^2_{\mu }(T^*M\otimes E)}+(\mathrm{d}\gamma \otimes w,\gamma \nabla z)_{L^2_{\mu }(T^*M\otimes E)}+(V\gamma w, \gamma z)_{\mu }. \end{aligned}$$

(5.4)

Setting $\xi :=d(\gamma ^2/2)\in T^*M$ and using equation (1.34) in Appendix C of [32] we have

$$\begin{aligned}&(\gamma \nabla w,\mathrm{d}\gamma \otimes z)_{L^2_{\mu }(T^*M\otimes E)}= (\nabla w, \xi \otimes z)_{L^2_{\mu }(T^*M\otimes E)}=(\nabla _{X} w, z)_{\mu }, \end{aligned}$$

(5.5)

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

$$\begin{aligned} (\mathrm{d}\gamma \otimes w,\gamma \nabla z)_{L^2_{\mu }(T^*M\otimes E)}&=(\xi \otimes w,\nabla z)_{L^2_{\mu }(T^*M\otimes E)} =(\nabla ^{*,\mu }(\xi \otimes w),z)_{\mu }\nonumber \\&=-({\text {div}}_{\mu }(X)w,z)_{\mu }-(\nabla _{X}w,z)_{\mu }, \end{aligned}$$

(5.6)

where, in local coordinates $x^{1},\,x^{2},\dots ,x^{n}$, for $X=X^j\frac{\partial }{\partial x^{j}}$, with Einstein summation convention,

$$\begin{aligned} {\text {div}}_{\mu }(X):=\frac{1}{\kappa }\left( \frac{\partial }{\partial x^{j}}\left( \kappa X^{j}\right) \right) . \end{aligned}$$

[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

$$\begin{aligned} {\text {div}}_{\mu }(X)=|\mathrm{d}\gamma |^2-\gamma (\Delta _{M,\mu }\gamma ), \end{aligned}$$

(5.7)

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

$$\begin{aligned} (\mathrm{d}\gamma \otimes w,\mathrm{d}\gamma \otimes z)_{L^2_{\mu }(T^*M\otimes E)}=\int _{M}|\mathrm{d}\gamma |^2\langle w, z\rangle \,\mathrm{d}\mu , \end{aligned}$$

we obtain

$$\begin{aligned} (H_1w,z)_{\mu _1}&=\int _{M}\langle \nabla w,\nabla z\rangle \gamma ^2\,\mathrm{d}\mu +\int _{M}\langle V w, z\rangle \gamma ^2\,\mathrm{d}\mu +\int _{M}\gamma (\Delta _{M,\mu }\gamma )\langle w,z\rangle \,\mathrm{d}\mu \nonumber \\&=(\nabla w,\nabla z)_{L^2_{\mu _1}(T^*M\otimes E)}+(Vw,z)_{\mu _1}+(\gamma ^{-1}(\Delta _{M,\mu }\gamma )w,z)_{\mu _1}\nonumber \\&=(\nabla ^{*,\mu _1}\nabla w, z)_{\mu _1}+(Vw,z)_{\mu _1}+(\gamma ^{-1}(\Delta _{M,\mu }\gamma )w,z)_{\mu _1}, \end{aligned}$$

(5.8)

which shows (5.3).

By (2.5) and (5.2) it follows that

$$\begin{aligned} \widetilde{V}(x)=\frac{\Delta _{M,\mu }\gamma }{\gamma }\,\text {Id}(x)+V(x)\ge \,(C-1)\text {Id}(x),\quad \text {for all }x\in M, \end{aligned}$$

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:

$$\begin{aligned} (D(\psi u),D(\psi u)) \ + \ (V\psi u, \psi u) \ = \ {\text {Re}}(\psi Hu,\psi u) \ + \ \Vert \widehat{D}(d\psi )u\Vert ^2. \end{aligned}$$

(6.1)

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

$$\begin{aligned} (\psi u, \,H (\psi u)) = \Vert \widehat{D}(d\psi )u\Vert ^2, \end{aligned}$$

(6.2)

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

$$\begin{aligned} (\psi u, \,H (\psi u)) = (\psi u, \, D^*D (\psi u)) \ + \ (V \psi u,\psi u) \ = \ (D(\psi u),D(\psi u)) \ + \ (V\psi u, \psi u), \end{aligned}$$

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 $u \in W_{\mathrm{comp}}^{1,2}(E)$,

$$\begin{aligned} (u,\, (H-\lambda ) u ) \ge \lambda _0^2\int _{M}\max \left( \dfrac{1}{r(x)^2},1 \right) |u(x)|^2\,\mathrm{d}\mu (x) + c_1 \Vert u\Vert ^2, \end{aligned}$$

(6.3)

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

$$\begin{aligned} F_{\varepsilon }(s)= \left\{ \begin{array}{ll} 0 &{}\quad \mathrm{for~} s\le \varepsilon \\ \rho (s-\varepsilon )/(\rho - \varepsilon ) &{}\quad \mathrm{for~} \varepsilon \le s \le \rho \\ s &{}\quad \mathrm{for~} \rho \le s \le 1 \\ 1 &{}\quad \mathrm{for~} 1 \le s \le R \\ R+1 -s &{}\quad \mathrm{for~} R \le s \le R+1 \\ 0 &{}\quad \mathrm{for~ } s \ge R+1. \end{array} \right. \end{aligned}$$

Let us fix $x_0\in M$. For any $\alpha >0$, we define the function $p_{\alpha }:M \rightarrow \mathbb {R}$ by

$$\begin{aligned} p_{\alpha }(x)=P_{\alpha }(d_{g^{\mathrm{TM}}}(x_0,x)), \end{aligned}$$

where $P_{\alpha }:[0,\infty ) \rightarrow \mathbb {R}$ is the continuous piecewise affine function defined by

$$\begin{aligned} P_{\alpha }(s)= \left\{ \begin{array}{ll} 1 &{}\quad \mathrm{for~} s\le 1/{\alpha } \\ -{\alpha } s + 2 &{}\quad \mathrm{for~} 1/{\alpha } \le s \le 2/{\alpha } \\ 0 &{}\quad \mathrm{for~ } s \ge 2/{\alpha }. \end{array} \right. \end{aligned}$$

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

$$\begin{aligned} \Vert \widehat{D}(\mathrm{d}(f_{\varepsilon }p_{\alpha }))v\Vert ^2\le \lambda _0^2\int _{M}|\mathrm{d}(f_{\varepsilon }p_{\alpha })(x)|^2|v(x)|^2\,\mathrm{d}\mu (x), \end{aligned}$$

(6.4)

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

$$\begin{aligned} (f_{\varepsilon }p_{\alpha } v,\, (H-\lambda ) (f_{\varepsilon }p_{\alpha } v)) \le \lambda _0^2\left( \frac{{\rho }}{{\rho -\varepsilon }}+\alpha \right) ^2\Vert v\Vert ^2. \end{aligned}$$

(6.5)

On the other hand, using the definitions of $f_{\varepsilon }$ and $p_{\alpha }$ and the assumption (6.3) we have

$$\begin{aligned} (f_{\varepsilon }p_{\alpha } v,\,(H-\lambda ) (f_{\varepsilon }p_{\alpha } v)) \ge \lambda _0^2\int _{S_{\rho ,R,\alpha }}|v(x)|^2\,\mathrm{d}\mu (x)+c_1 \Vert f_{\varepsilon }p_{\alpha } v \Vert ^2, \end{aligned}$$

(6.6)

where

$$\begin{aligned} S_{\rho ,R,\alpha }:=\{x\in M:\rho \le r(x) \le R \text { and } d_{g^{\mathrm{TM}}}(x_0,x) \le 1/\alpha \}. \end{aligned}$$

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

$$\begin{aligned} \lambda _0^2\int _{S_{\rho ,R,\alpha }}|v(x)|^2\,\mathrm{d}\mu (x) \ + \ c_1 \Vert f_{\varepsilon }p_{\alpha } v\Vert ^2\le \lambda _0^2\left( \frac{{\rho }}{{\rho -\varepsilon }}+\alpha \right) ^2\Vert v\Vert ^2. \end{aligned}$$

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

$$\begin{aligned} (u,\, Hu) \ = \ (u,D^*Du) \ + \ (Vu,u) \ = \ (Du,Du) \\ + \ (Vu,u) \ \ge \ (Vu,u), \quad \text { for all }u\in W_{\mathrm{comp}}^{1,2}(E), \end{aligned}$$

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:

$$\begin{aligned} (u,\, (H-\lambda ) u)&\ge \lambda _0^2\int _{M}\dfrac{1}{r(x)^2}|u(x)|^2\,\mathrm{d}\mu (x)-(\lambda +C)\Vert u\Vert ^2\nonumber \\&\ge \lambda _0^2\int _{M}\max \left( \dfrac{1}{r(x)^2},1 \right) |u(x)|^2\,\mathrm{d}\mu (x) -(\lambda +C+1)\Vert u\Vert ^2. \end{aligned}$$

(6.7)

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. $\square$