Localization principles for Schr\"odinger operator with a singular matrix potential

TL;DR

This paper extends localization principles to the spectrum of one-dimensional Schrödinger operators with matrix singular potentials, providing conditions for the spectrum to be bounded below and discrete.

Contribution

It generalizes Ismagilov's localization principles to matrix singular potentials, offering new criteria for spectral properties of such operators.

Findings

Conditions for spectrum to be bounded below

Criteria for spectrum to be discrete

Generalizations of localization principles

Abstract

We study the spectrum of the one-dimensional Schr\"{o}dinger operator $H_0$ with a matrix singular distributional potential $q=Q'$ where $Q\in L^{2}_{\mathrm{loc}}(\mathbb{R},\mathbb{C}^{m})$. We obtain generalizations of Ismagilov's localization principles, which give necessary and sufficient conditions for the spectrum of $H_0$ to be bounded below and discrete.