On the Bands of the Schrodinger Operator with a Matrix Potential

TL;DR

This paper studies the spectral bands and gaps of a one-dimensional Schrödinger operator with a matrix potential, establishing conditions for the number of spectral gaps and the overlap of bands on the positive real axis.

Contribution

It provides new insights into the spectral structure of matrix potential Schrödinger operators, including conditions for finite gaps and band overlap.

Findings

Main part of positive real axis overlapped by m bands

Condition for finite number of spectral gaps

Spectral properties depend on the matrix potential Q

Abstract

In this article we consider the one-dimensional Schrodinger operator L(Q) with a Hermitian periodic m by m matrix potential Q. We investigate the bands and gaps of the spectrum and prove that the main part of the positive real axis is overlapped by m bands. Moreover, we find a condition on the potential Q for which the number of gaps in the spectrum of L(Q) is finite.