Schr\"{o}dinger operators with matrix potentials. Transition from the absolutely continuous to the singular spectrum

TL;DR

This paper proves that matrix Schrödinger operators with square-integrable or smooth decaying potentials have an absolutely continuous spectrum identical to the unperturbed operator, highlighting spectral stability under certain conditions.

Contribution

It establishes the spectral equivalence for matrix Schrödinger operators with specific classes of potentials, extending known results to matrix and smoother potential cases.

Findings

Absolutely continuous spectrum matches unperturbed operator for square-integrable potentials.

Results also hold for some slowly decaying smooth potentials.

Spectral transition from absolutely continuous to singular spectrum is characterized.

Abstract

It is proven that the absolutely continuous spectrum of matrix Schr\"{o}dinger operators coincides (with the multiplicity taken into account) with the spectrum of the unperturbed operator if the (matrix) potential is square integrable. The same result is also proven for some classes of slower decaying potentials if they are smooth.