Conditions for discreteness of the spectrum to Schr\"odinger operator via non-increasing rearrangement, Lagrangian relaxation and perturbations

TL;DR

This paper develops new criteria for the discreteness of the spectrum of Schrödinger operators using rearrangement, Lagrangian relaxation, and perturbation techniques, extending previous results to more general potential behaviors.

Contribution

It introduces novel sufficient conditions for spectrum discreteness based on potential rearrangement, expectation, deviation, and perturbations, broadening the applicability of prior criteria.

Findings

Conditions in terms of potential rearrangement established

Discreteness criteria derived via Lagrangian relaxation methods

Results extend to potentials with localized infinity sets

Abstract

This work is a continuation of our previos paper, where for the Schr\"odinger operator $H=-\Delta+ V(\e)\cdot$ $(V(\e)\ge 0)$, acting in the space $L_2(\R^d)\,(d\ge 3)$, some sufficient conditions for discreteness of its spectrum have been obtained on the base of well known Mazya -Shubin criterion and an optimization problem for a set function, which is an infinite-dimensional generalization of a binary linear programming problem. A sufficient condition for discreteness of the spectrum is formulated in terms of the non-increasing rearrangement of the potential $V(\e)$. Using the method of Lagrangian relaxation for this optimization problem, we obtain a sufficient condition for discreteness of the spectrum in terms of expectation and deviation of the potential. By means of suitable perturbations of the potential we obtain conditions for discreteness of the spectrum, covering potentials which tend to infinity only on subsets of cubes, whose Lebesgue measures tend to zero when the cubes go to infinity. Also the case where the operator $H$ is defined in the space $L_2(\Omega)$ is considered ($\Omega$ is an open domain in $\R^d$).