Self-adjoint, globally defined Hamiltonian operators for systems with boundaries

TL;DR

This paper characterizes all self-adjoint Hamiltonians that confine quantum systems to a domain, explicitly constructing boundary potentials for systems like the Schrödinger operator, with applications to deformation quantization.

Contribution

It provides a complete description of self-adjoint Hamiltonians with boundary confinement, including explicit boundary potential constructions for Schrödinger operators.

Findings

Explicit boundary potential forms for confining Hamiltonians

Construction of self-adjoint operators on maximal domains

Application to deformation quantization of boundary systems

Abstract

For a general self-adjoint Hamiltonian operator $H_0$ on the Hilbert space $L^2(\RE^d)$, we determine the set of all self-adjoint Hamiltonians $H$ on $L^2(\RE^d)$ that dynamically confine the system to an open set $\Omega \subset \RE^d$ while reproducing the action of $ H_0$ on an appropriate operator domain. In the case $H_0=-\Delta +V$ we construct these Hamiltonians explicitly showing that they can be written in the form $H=H_0+ B$, where $B$ is a singular boundary potential and $H$ is self-adjoint on its maximal domain. An application to the deformation quantization of one-dimensional systems with boundaries is also presented.