Surgery transformations and spectral estimates of $\delta$ beam operators

TL;DR

This paper introduces delta-type vertex conditions for beam operators on metric graphs, explores how graph modifications affect spectra, and develops surgery principles to estimate eigenvalues, extending quantum graph techniques to beam operators.

Contribution

It develops a framework for delta-type conditions for beam operators and establishes surgery principles to analyze spectral changes due to graph modifications.

Findings

Established monotonicity properties of spectra under graph surgery.

Derived lower and upper bounds for eigenvalues of beam operators.

Extended delta vertex conditions from quantum graphs to beam operators.

Abstract

We introduce $\delta$ type vertex conditions for beam operators, the fourth derivative operator, on metric graphs and study the effect of certain geometrical alterations (graph surgery) of the graph on the spectra of beam operators on compact metric graphs. Results are obtained for a class of vertex conditions which can be seen as an analogue of {\delta} vertex conditions for quantum graphs. There are a number of possible candidates of {\delta} type conditions for beam operators. We develop surgery principles and record the monotonicity properties of the spectrum, keeping in view the possibility that vertex conditions may change within the same class after certain graph alterations. We also demonstrate the applications of surgery principles by obtaining several lower and upper estimates on the eigenvalues.