Counting spectrum via the Maslov index for one dimensional $\theta-$periodic Schr\"odinger operators

TL;DR

This paper investigates the spectrum of matrix-valued Schr"odinger operators with $ heta$-periodic boundary conditions, using the Maslov index to relate eigenvalue counts and derivatives, and provides a new proof connecting Morse and Maslov indices.

Contribution

It introduces a novel method to compute eigenvalue differences and derivatives for $ heta$-periodic Schr"odinger operators using the Maslov index, and offers a shorter proof of a recent index relation.

Findings

Derived a formula for eigenvalue differences using the Maslov index.

Established a relation between eigenvalue derivatives and the Maslov crossing form.

Provided a new, shorter proof connecting Morse and Maslov indices.

Abstract

We study the spectrum of the Schr\"odinger operators with $n\times n$ matrix valued potentials on a finite interval subject to $\theta-$periodic boundary conditions. For two such operators, corresponding to different values of $\theta$, we compute the difference of their eigenvalue counting functions via the Maslov index of a path of Lagrangian planes. In addition we derive a formula for the derivatives of the eigenvalues with respect to $\theta$ in terms of the Maslov crossing form. Finally, we give a new shorter proof of a recent result relating the Morse and Maslov indices of the Schr\"odinger operator for a fixed $\theta$.