Electron. J. Diff. Eqns., Vol. 2003(2003), No. 64, pp. 1-8.

Self-adjointness of Schrodinger-type operators with singular potentials on manifolds of bounded geometry

Ognjen Milatovic

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 $M$, $b greater than 0$ 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

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Ognjen Milatovic
78 Apsley Street, Apt. 1
Hudson, MA 01749, USA
email: omilatov@unf.edu

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