On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in R^n

Abstract

Let $\Omega$ be a bounded domain in $R^n$ with $C^2$-smooth boundary of co-dimension 1, and let $H=-\Delta +V(x)$ be a Schrödinger operator on $\Omega$ with potential V locally bounded. We seek the weakest conditions we can find on the rate of growth of the potential V close to the boundary which guarantee essential self-adjointness of H on $C_0^{\infty }(\Omega )$. As a special case of an abstract condition, we add optimal logarithmic type corrections to the known condition $V(x)\ge \frac{3}{4d(x{)}^2}$, where $d(x)=dist(x,\partial \Omega )$. The constant 1 in front of each logarithmic term in Theorem 2 is optimal. The proof is based on a refined Agmon exponential estimate combined with a well known multidimensional Hardy inequality.

Cited by in corpus (11)

• Essential self-adjointness for combinatorial Schrödinger operators II- Metrically non complete graphs
• Laplaciens de graphes infinis I Graphes métriquement complets
• On the essential self-adjointness of singular sub-Laplacians
• On Geometric Quantum Confinement in Grushin-type Manifolds
• Quantum Geometric Confinement and Dynamical Transmission in Grushin Cylinder
• Weyl's law for singular Riemannian manifolds
• Quantum particle across Grushin singularity
• Point interactions for 3D sub-Laplacians
• Results on the Aharonov-Bohm effect without contact with the solenoid
• On essential self-adjointness for magnetic Schroedinger and Pauli operators on the unit disc in R^2
• Essential self-adjointness of symmetric first-order differential systems and confinement of Dirac particles on bounded domains in ${\mathbb{R}}^d$