Surgery transformations and eigenvalue estimates for quantum graphs with $\delta'$ vertex interactions

TL;DR

This paper develops new methods for analyzing quantum graphs with $ abla'$ interactions, providing bounds on eigenvalues by extending surgical techniques and exploring eigenvalue monotonicity.

Contribution

It introduces extended surgery tools for quantum graphs with $ abla'$ conditions and establishes eigenvalue bounds based on vertex parameter signs.

Findings

Eigenvalue bounds for $ abla'$ and $ abla$ Laplacians.

Monotonicity properties depend on vertex parameter signs.

Interlacing inequalities between different vertex condition eigenvalues.

Abstract

We extend the surgical tool box for quantum graphs to anti-standard and $\delta'$ vertex conditions. Monotonicity properties of eigenvalues of graph Laplacian with $\delta'$ interactions at vertices depend on the sign of vertex parameter. Using several interlacing inequalities between eigenvalues of graph Laplacian with different vertex conditions and surgery principles we obtain new upper and lower bounds on the eigenvalues of $\delta$ and $\delta'$ Laplacians.