Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 64, pp. 1-8.
Title: Self-adjointness of Schrodinger-type operators
with singular potentials on manifolds of bounded geometry
Author: Ognjen Milatovic (Hudson, MA, USA)
Abstract:
We consider the Schrodinger type differential expression
$$ H_V=\nabla^*\nabla+V, $$
where $\nabla$ is a $C^{\infty}$-bounded
Hermitian connection on a Hermitian vector bundle $E$ of bounded
geometry over a manifold of bounded geometry $(M,g)$ with metric
$g$ and positive $C^{\infty}$-bounded measure $d\mu$, and
$V=V_1+V_2$, where $0\leq V_1\in L_{\rm loc}^1(\mathop{\rm End} E)$
and $0\geq V_2\in L_{\rm loc}^1(\mathop{\rm End} E)$ are linear
self-adjoint bundle endomorphisms. We give a sufficient condition
for self-adjointness of the operator $S$ in $L^2(E)$ defined by
$Su=H_Vu$ for all $u\in\mathop{\rm Dom}(S)=\{u\in
W^{1,2}(E)\colon \int\langle V_1u,u\rangle\,d\mu<+\infty
\hbox{ and }H_Vu\in L^2(E)\}$.
The proof follows the scheme of Kato, but
it requires the use of more general version of Kato's inequality
for Bochner Laplacian operator as well as a result on the
positivity of $u\in L^2(M)$ satisfying the equation
$(\Delta_M+b)u=\nu$, where $\Delta_M$ is the scalar Laplacian on
$M$, $b>0$ is a constant and $\nu\geq 0$ is a positive
distribution on $M$.
Submitted May 13, 2003. Published June 11, 2003.
Math Subject Classifications: 35P05, 58J50, 47B25, 81Q10.
Key Words: Schrodinger operator; self-adjointness; manifold;
bounded geometry; singular potential