# Conditions for discreteness of the spectrum to multi-dimensional Schr\"odinger operator

## TL;DR

This paper extends previous work on conditions ensuring the discreteness of the spectrum of multi-dimensional Schrödinger operators, introducing more general criteria based on harmonic capacity and rearrangement techniques.

## Contribution

It develops new sufficient conditions for spectrum discreteness using capacitary inequalities, base polyhedron concepts, and rearrangement methods, generalizing earlier criteria.

## Key findings

- Derived more general spectrum discreteness conditions
- Utilized capacitary strong inequalities and base polyhedron theory
- Applied rearrangement techniques to potential functions

## Abstract

This work is a continuation of our previos paper \cite{Zel1}, where for the 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 constructive 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. Using a {\it capacitary strong type inequality} of David Adams, the concept of {\it base polyhedron} for the harmonic capacity and some properties of Choquet integral by this capacity, we obtain more general sufficient conditions for discreteness of the spectrum of $H$ in terms of a repeated nonincreasing rearrangement of the function $Y(\e,\bt)=\sqrt{V(\e)}\frac{1}{|\e-\bt|^{d-2}}\sqrt{V(\bt)}$ on cubes that are going to infinity.