Spectral Zeta Functions for a Cylinder and a Circle

TL;DR

This paper constructs explicit spectral zeta functions for scalar fields on a cylinder and circle, enabling calculation of Casimir energies for these geometries with boundary conditions.

Contribution

It provides explicit formulas for spectral zeta functions on cylindrical and circular geometries, simplifying Casimir energy computations without extra calculations.

Findings

Spectral zeta functions are explicitly constructed for a cylinder.

Spectral zeta functions for a circle are derived from the cylindrical case.

Casimir energies are calculated for the given boundary conditions.

Abstract

Spectral zeta functions $\zeta(s)$ for the massless scalar fields obeying the Dirichlet and Neumann boundary conditions on a surface of an infinite cylinder are constructed. These functions are defined explicitly in a finite domain of the complex plane s containing the closed interval of real axis $-1\le$ Re $s \le 0$. Proceeding from this the spectral zeta functions for the boundary conditions given on a circle (boundary value problem on a plane) are obtained without any additional calculations. The Casimir energy for the relevant field configurations is deduced.