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

Milatovic, Ognjen

Electronic journal of differential equations, Tome 2003 (2003)

Résumé

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 $M$.

Détail

Classification : 35P05, 58J50, 47B25, 81Q10
Keywords: Schrödinger operator, self-adjointness, manifold, bounded geometry