Spectral monotonicity for Schr\"odinger operators on metric graphs

TL;DR

This paper investigates how geometric changes to compact metric graphs affect the spectra of Schrödinger operators, revealing that adding edges or connecting vertices influences eigenvalues monotonically, with effects depending on vertex condition signs.

Contribution

It extends spectral monotonicity results to permutation invariant vertex conditions, including δ and δ'-type, highlighting differences from classical Kirchhoff conditions.

Findings

Adding edges or joining vertices causes eigenvalues to change monotonically.

Monotonicity properties vary with vertex condition signs.

Results differ from classical Kirchhoff and δ-condition cases.

Abstract

We study the influence of certain geometric perturbations on the spectra of self-adjoint Schr\"odinger operators on compact metric graphs. Results are obtained for permutation invariant vertex conditions, which, amongst others, include $\delta$ and $\delta'$-type conditions. We show that adding edges to the graph or joining vertices changes the eigenvalues monotonically. However, the monotonicity properties may differ from what is known for the previously studied cases of Kirchhoff (or standard) and $\delta$-conditions and may depend on the signs of the coefficients in the vertex conditions.