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Relative-Zeta and Casimir Energy for a Semitransparent Hyperplane Selecting Transverse Modes

  • Claudio CacciapuotiEmail author
  • Davide Fermi
  • Andrea Posilicano
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 18)

Abstract

We study the relative zeta function for the couple of operators A0 and A α , where A0 is the free unconstrained Laplacian in L2(R d ) (d ≥ 2) and A α is the singular perturbation of A0 associated to the presence of a delta interaction supported by a hyperplane. In our setting the operatorial parameter α, which is related to the strength of the perturbation, is of the kind α = α(−Δ), where −Δ is the free Laplacian in L2(R d−1). Thus α may depend on the components of the wave vector parallel to the hyperplane; in this sense A α describes a semitransparent hyperplane selecting transverse modes.

As an application we give an expression for the associated thermal Casimir energy. Whenever α = χ I (−Δ), where χ I is the characteristic function of an interval I, the thermal Casimir energy can be explicitly computed.

Keywords

Casimir effect Delta-interactions Finite temperature quantum fields Relative zeta function Zeta regularization 

MSC 2010

81Q10 81T55 81T10 

Notes

Acknowledgements

C.C. and D.F. acknowledge the support of the FIR 2013 project “Condensed Matter in Mathematical Physics”, Ministry of University and Research of Italian Republic (code RBFR13WAET).

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Claudio Cacciapuoti
    • 1
    Email author
  • Davide Fermi
    • 2
  • Andrea Posilicano
    • 1
  1. 1.DiSAT, Sezione di MatematicaUniversità dell’InsubriaComoItaly
  2. 2.Dipartimento di MatematicaUniversità di MilanoMilanoItaly

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