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

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 $\R^d$).