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Vacuum energy of scalar fields on spherical shells with general matching conditions

Guglielmo Fucci, César Romaniega Sancho

Abstract

In this work we analyze the spectral zeta function for massless scalar fields propagating in a $D$-dimensional flat space under the influence of a shell potential. The shell potential is defined in terms of the two-interval self-adjoint extensions of the Hamiltonian describing the dynamics of the scalar field. After performing the necessary analytic continuation, we utilize the spectral zeta function of the system to compute the vacuum energy of the field.

Vacuum energy of scalar fields on spherical shells with general matching conditions

Abstract

In this work we analyze the spectral zeta function for massless scalar fields propagating in a -dimensional flat space under the influence of a shell potential. The shell potential is defined in terms of the two-interval self-adjoint extensions of the Hamiltonian describing the dynamics of the scalar field. After performing the necessary analytic continuation, we utilize the spectral zeta function of the system to compute the vacuum energy of the field.
Paper Structure (15 sections, 113 equations, 2 figures)

This paper contains 15 sections, 113 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The shell potential
  3. The spectral zeta function
  4. Analytic continuation of the spectral zeta function
  5. Case $b\neq 0$.
  6. Case $b=0$.
  7. Casimir energy
  8. Case $b\neq 0$.
  9. Case $b=0$.
  10. Conclusions
  11. On the self-adjoint extensions of the radial operator
  12. Uniform asymptotic expansion of the characteristic function ${\cal F}_{\nu}(\nu z)$
  13. Case $b\neq 0$.
  14. Case $b=0$.
  15. An identity for some generalized Bernoulli polynomials

Figures (2)

  • Figure 1: Plot of the Casimir energy $E_{\textrm{Cas}}$ for $R=1$ as a function of the parameter $a$ when $b>(a^2+1)R/a$. The graphs are labeled according to the value of the parameter $b$ used for evaluation of the energy.
  • Figure 2: Plot of the Casimir energy $E_{\textrm{Cas}}$ for $R=1$ as a function of the parameter $d$ when $c$ satisfies (\ref{['equationzero']}).

Theorems & Definitions (1)

  • Definition 2.1