Minimal potential results for the Schrodinger equation in a slab

TL;DR

This paper establishes necessary conditions for solutions of the Schrödinger equation in a slab with bounded potential, linking potential magnitude, support measure, and spectral parameters, with many conditions proven to be sharp.

Contribution

It provides new necessary conditions for the existence of solutions to the Schrödinger equation in a slab, involving potential bounds and geometric constraints, many of which are sharp.

Findings

Necessary conditions relate potential size, support measure, and spectral distance.

Conditions are sharp in many cases.

Results extend understanding of Schrödinger solutions in constrained geometries.

Abstract

Consider the Schrodinger equation -\Delta u =(k+V) u in an infinite slab S= \R^{n-1}x (0,1), where V is a bounded potential supported on a set D of finite measure. We prove necessary conditions for the existence of nontrivial admissible solutions. These conditions involve the sup. of |V|, the measure of D, and the distance of k from the "special set" {\pi^2 m^2, m positive integer}. In many cases, these inequalities are sharp.