$\zeta$-function for a model with spectral dependent boundary conditions

TL;DR

This paper analyzes the meromorphic structure of the $ta$-function for a Sturm-Liouville operator with spectral dependent boundary conditions, deriving poles, and applying results to compute determinants and Casimir energy.

Contribution

It provides a detailed study of the $ta$-function's poles for operators with spectral boundary conditions and applies this to physical quantities like determinants and Casimir energy.

Findings

The $ta$-function has simple poles following standard rules.

Explicit formulas for the determinant and Casimir energy are derived.

Dependence of these quantities on the segment length $l$ is analyzed.

Abstract

We explore the meromorphic structure of the $\zeta$-function associated to the boundary eigenvalue problem of a modified Sturm-Liouville operator subject to spectral dependent boundary conditions at one end of a segment of length $l$. We find that it presents isolated simple poles which follow the general rule valid for second order differential operators subject to standard local boundary conditions. We employ our results to evaluate the determinant of the operator and the Casimir energy of the system it describes, and study its dependence on $l$ for both the massive and the massless cases.