Path integrals and the essential self-adjointness of differential operators on noncompact manifolds

Abstract

We consider Schrödinger operators on possibly noncompact Riemannian manifolds, acting on sections in vector bundles, with locally square integrable potentials whose negative part is in the underlying Kato class. Using path integral methods, we prove that under geodesic completeness these differential operators are essentially self-adjoint on $\mathsf{C }^{\infty }_0$, and that the corresponding operator closures are semibounded from below. These results apply to nonrelativistic Pauli–Dirac operators that describe the energy of Hydrogen type atoms on Riemannian $3$-manifolds.

Appendices

Appendix A. Friedrichs mollifiers

We record the following result on Friedrichs mollifiers here. Let $M$ be a smooth connected Riemannian manifold without boundary, $E\rightarrow M$ a smooth Hermitian vector bundle, $\nabla$ a Hermitian covariant derivative in $E$, and $V :M\rightarrow \mathrm{End}(E)$ a potential.

Proposition A.1

Let $|V|\in \mathsf{L }^2_\mathrm{loc }(M)$ and assume that $f\in \Gamma _\mathsf{L ^{\infty }_\mathrm{loc }}(M,E)$ is compactly supported with $\nabla ^{\dagger }\nabla f\in \Gamma _{ \mathsf{L }^{2}_\mathrm{loc }}(M,E)$ in the sense of distributions. Then there is a sequence $(f_n)_{n\in \mathbb{N }}\subset \Gamma _{\mathsf{C }^{\infty }_0}(M,E)$ such that

$$\begin{aligned}&\lim _{n\rightarrow \infty }\Vert f_n-f\Vert =0,\nonumber \\&\lim _{n\rightarrow \infty }\Vert \nabla ^{\dagger }\nabla f_n -\nabla ^{\dagger }\nabla f\Vert =0,\nonumber \\&\lim _{n\rightarrow \infty }\Vert V f_n -Vf\Vert =0.\nonumber \end{aligned}$$

Remark A.2

Note that one indeed has $f\in \Gamma _{ \mathsf L ^{2}}(M,E)$, which follows from $f\in \Gamma _{ \mathsf L ^{\infty }_\mathrm{loc }}(M,E)$ and the fact that $f$ has a compact support. Furthermore, $\nabla ^{\dagger }\nabla f\in \Gamma _{ \mathsf L ^{2}}(M,E)$ follows from $\nabla ^{\dagger }\nabla f\in \Gamma _{ \mathsf L ^{2}_\mathrm{loc }}(M,E)$ and the fact that $\nabla ^{\dagger }\nabla f$ has a compact support.

Proof of Proposition 4.3

Since most of the arguments should be well-known, we only sketch the proof. Let $m:=\dim M$ and let $d$ be the fiber dimension of $E$. Since $f$ is compactly supported, we can use a partition of unity argument to assume that $f$ is supported in a relatively compact coordinate domain $U\subset M$ (which is identified with an open subset of $\mathbb{R }^m$) such that there is a smooth orthonormal frame for $E$ over $U$, and we denote the components of $f$ in this frame with $f^{(1)},\ldots ,f^{(d)}$. Now take some $0\le j_r\in \mathsf{C }^{\infty }_0(\mathbb{R }^m)$ with $j(z)=0$ for $|z|\ge 1$ and

$$\begin{aligned} \int _{\mathbb{R }^m}j(z)\mathrm{d}z=1. \end{aligned}$$

For $r>0$ let $j_r\in \mathsf{C }^{\infty }_0(\mathbb{R }^m)$ be given by $j_r(z)=r^{-m}j(r^{-1} z)$. Let $r>0$ be small enough in the following such that the functions

$$\begin{aligned} x\longmapsto \int _{\mathbb{R }^m} j_r(x-y)f^{(i)}(y) \mathrm{d}y,\quad i=1,\dots , d, \end{aligned}$$

(46)

define an element

$$\begin{aligned} f_r\in \Gamma _{\mathsf{C }^{\infty }_0}(U,E)\subset \Gamma _{\mathsf{C }^{\infty }_0}(M,E). \end{aligned}$$

Since the sections $f_r-f$ and $\nabla ^{\dagger }\nabla f_r -\nabla ^{\dagger }\nabla f$ are compactly supported, the convergence

$$\begin{aligned} \lim _{r\rightarrow 0+}\Vert f_r-f\Vert =0 \end{aligned}$$

(47)

follows from Lemma 5.13 (ii) in [1], and

$$\begin{aligned} \lim _{r\rightarrow 0+}\Vert \nabla ^{\dagger }\nabla f_r -\nabla ^{\dagger }\nabla f\Vert =0 \end{aligned}$$

follows from the $\mathsf{L }^2_\mathrm{loc }$-version of Proposition 5.14 in [1], which can be proven with analogous arguments. Note that so far we have only used that $f$ is locally square integrable with a compact support.

The local boundedness assumption on $f$ comes into play as follows: Namely, this assumption combined with the compact support assumption implies that $f$ is actually bounded and so (46) implies

$$\begin{aligned} |f_r(x)|_x\le \Vert f\Vert _{\infty }\quad \text{ for} \text{ all}\; x, r. \end{aligned}$$

(48)

Since [in view of (47)] we may assume that $f_r\rightarrow f$ a.e. in $M$, and since $f_r$ has a compact support, the required convergence

$$\begin{aligned} \lim _{r\rightarrow 0+}\Vert V f_r -V f \Vert =0 \end{aligned}$$

now follows from (48) and dominated convergence. $\square$

Appendix B. Finite speed of propagation

The following lemma is usually referred to as Chernoff’s finite speed of propagation method [2]. Let $M$ be a smooth connected Riemannian manifold without boundary, and let $E\rightarrow M$ be a smooth Hermitian vector bundle.

Lemma B.1

Let $S$ be a self-adjoint nonnegative operator in $\Gamma _\mathsf{L ^{2}}(M,E)$. Assume furthermore that $\mathsf{D }^0(S)$, the compactly supported elements of $\mathsf{D }(S)$, are dense in $\Gamma _\mathsf{L ^{2}}(M,E)$ and that for any $f\in \mathsf{D }^0(S)$ and any $t>0$, the section $\cos (t\sqrt{S})f$ has a compact support. Then $\mathsf{D }^0(S)$ is an operator core for $S$.

Proof

The proof is a straightforward generalisation of the proof of Theorem 3 in [13]. $\square$

Appendix C. Path ordered exponentials

In the following lemma, we collect some known facts about path ordered exponentials for the convenience of the reader:

Lemma 6.1

Let ${\fancyscript{H}}$ be a finite dimensional Hilbert space, let $T\in (0,\infty ]$ and let $F\in \mathsf L ^1_\mathrm{loc }([0,T),{\fancyscript{L}}({\fancyscript{H}}))$. Then the following assertions hold:

• (a) There is a unique weak ($=\mathsf{AC }_\mathrm{loc }$) solution $Y:[0,T)\rightarrow {\fancyscript{L}}({\fancyscript{H}})$ of the ordinary initial value problem

  $$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} Y(t)=Y(t)F(t),Y(0)=\mathbf 1 . \end{aligned}$$

  (49)

• (b) For any $0\le t<T$ one has

  $$\begin{aligned} Y(t)=\mathbf 1 +\sum ^{\infty }_{k=1}\,\,\,\int _{0\le s_1\le \ldots \le s_k\le t} F(s_1)\ldots F(s_k) \mathrm{d}s_1\dots \mathrm{d}s_k. \end{aligned}$$

  (50)

• (c) If $F(\bullet )$ is Hermitian a.e. in $[0,T)$ and if there exists a real-valued function $c\in \mathsf L ^1_\mathrm{loc }[0,T)$ such that for all $v\in {\fancyscript{H}}$ it holds that

  $$\begin{aligned} \langle F(\bullet ) v, v\rangle _{{\fancyscript{H}}}\le c(\bullet ) \Vert v\Vert ^2_{{\fancyscript{H}}\quad }\text{ a.e.} \text{ in}\; [0,T), \end{aligned}$$

  then one has

  $$\begin{aligned} \Vert Y(t)\Vert _{{\fancyscript{H}}}\le \text{ e}^{\int ^t_0 c(s) \mathrm{d}s}\quad \mathrm{for\;all}\; 0\le t<T. \end{aligned}$$

Proof

See [7] and the Appendix C of [15]. $\square$