Conditions of discreteness of the spectrum for Schr\"odinger operator and some optimization problems for capacity and measures

TL;DR

This paper provides constructive conditions for the discreteness of the spectrum of Schrödinger operators in higher dimensions, using capacity-based optimization and measure rearrangements, with applications to spectral theory and potential analysis.

Contribution

It introduces a new optimization framework involving harmonic capacity and measures to establish spectral discreteness criteria for Schrödinger operators, extending previous criteria.

Findings

Derived sufficient conditions for spectrum discreteness using measure-based rearrangements.

Solved an infinite-dimensional optimization problem related to capacity and measures.

Constructed counterexamples to compare with existing spectral criteria.

Abstract

For the the Schr\"odinger operator $H=-\Delta+ V(x)\cdot$, acting in the space L_2(\R^d)\,(d\ge 3), with V(x)\ge 0 and V(\cdot)\in L_{1,loc}(\R^d), we obtain some constructive conditions for discreteness of its spectrum. Basing on the Mazya-Shubin criterion for discreteness of the spectrum of $H$ and using the isocapacity inequality and the concept of base polyhedron for the harmonic capacity, we have estimated from below the cost functional of an optimization problem, involved in this criterion, replacing a submodular constrain (in terms of the harmonic capacity) by a weaker but additive constrain (in terms of a measure). By this way we obtain an optimization problem, which can be considered as an infinite-dimensional analogue of the optimal covering problem. We have solved this problem for the case of a non-atomic measure. This approach enables us to obtain for the operator H some sufficient conditions for discreteness of its spectrum in terms of non-increasing rearrangements, with respect to measures from the base polyhedron, for some functions connected with the potential V(x). We construct some counterexamples, which permit to compare our results between themselves and with results of other authors.