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$ζ$-function for a model with spectral dependent boundary conditions

H. Falomir, M. Loewe, E. Muñoz, J. C. Rojas

TL;DR

This work analyzes a scalar field on a finite segment under dynamical boundary conditions at one end, leading to a boundary eigenvalue problem with spectral-parameter dependence. By deriving the spectrum and employing contour-integral representations, the authors establish the meromorphic extension of the associated $\zeta$-function $\zeta_A(s)$, showing isolated simple poles at half-integer values, consistent with Seeley-type results for second-order operators. They extract the functional determinant via $\log\mathrm{Det}(A/\mu^2)=-\zeta_A'(0)$ and compute the Casimir energy $E_{Cas}(l)$, revealing renormalization effects that include a linear energy density term in $l$ for large $l$, and mass-dependent finite parts via integral representations. The analysis covers both massive and massless cases, providing explicit expressions and asymptotic behaviors for $E_{Cas}(l)$ and $\mathrm{Det}(A)$, with clear dependence on boundary-parameter choices and their renormalization-scale $\mu$. The results have implications for quantum field theories with dynamical boundaries, including potential experimental and holographic contexts, and demonstrate robust methods for handling spectral-parameter boundary conditions in Casimir-type settings.

Abstract

We explore the meromorphic structure of the $ζ$-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.

$ζ$-function for a model with spectral dependent boundary conditions

TL;DR

This work analyzes a scalar field on a finite segment under dynamical boundary conditions at one end, leading to a boundary eigenvalue problem with spectral-parameter dependence. By deriving the spectrum and employing contour-integral representations, the authors establish the meromorphic extension of the associated -function , showing isolated simple poles at half-integer values, consistent with Seeley-type results for second-order operators. They extract the functional determinant via and compute the Casimir energy , revealing renormalization effects that include a linear energy density term in for large , and mass-dependent finite parts via integral representations. The analysis covers both massive and massless cases, providing explicit expressions and asymptotic behaviors for and , with clear dependence on boundary-parameter choices and their renormalization-scale . The results have implications for quantum field theories with dynamical boundaries, including potential experimental and holographic contexts, and demonstrate robust methods for handling spectral-parameter boundary conditions in Casimir-type settings.

Abstract

We explore the meromorphic structure of the -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 . 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 for both the massive and the massless cases.

Paper Structure

This paper contains 21 sections, 155 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The dynamical equation
  3. The spectrum
  4. Alternative evaluation of the meromorphic extension of $\zeta_A(s)$ for the massive case
  5. The determinant of $A$
  6. The Casimir energy
  7. The Casimir energy as a function of $l$
  8. The massless case
  9. The determinant of $A$ for the massless case
  10. The Casimir energy for the massless case
  11. Conclusions
  12. Iterative evaluation of $\zeta_A(s)$ for the $\beta'_2=0$ case
  13. Bounds of $E_{Cas}(l)$ for small $l$ with $m=0$
  14. The contribution of $I_2(l)$ for $\mu l\ll 1$
  15. $\lim_{y \rightarrow\infty}{q(y)}=-1$ ($\beta'_2\neq 0$)
  16. ...and 6 more sections

Figures (10)

  • Figure 1: The spectrum ($l=1,a=-1,b=7,c=-3,d=-4$).
  • Figure 2: $\Delta E$ as a function of $M=m l$, with $b_1=1/2, b_2=-1, b'_1=-1/2, b'_2=1/2$.
  • Figure 3: $\Delta E$ as a function of $M=m l$ for $b_1=1, b_2=-150, b'_1=-130, b'_2=7$.
  • Figure 4: $\Delta E$ as a function of $M= m l$ for $b_1=10, b_2=-22, b'_1=-1, b'_2=0$.
  • Figure 5: $\Delta E$ as a function of $M= m l$ for $b_1=61, b_2=-23, b'_1=-1, b'_2=0$.
  • ...and 5 more figures