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2017 | OriginalPaper | Chapter

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

Authors : Claudio Cacciapuoti, Davide Fermi, Andrea Posilicano

Published in: Advances in Quantum Mechanics

Publisher: Springer International Publishing

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Abstract

We study the relative zeta function for the couple of operators A 0 and A α , where A 0 is the free unconstrained Laplacian in L 2(R d ) (d ≥ 2) and A α is the singular perturbation of A 0 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 L 2(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.

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Appendix
Available only for authorised users
Footnotes
1
We remark that here we are slightly abusing terminology, as “relative spectral measure” usually denotes the function \(e(v):={ v \over i\pi } \lim _{\varepsilon \rightarrow 0^{+}}\mathop{ \mathrm{Tr}}\nolimits \,(R(v^{2} + i\varepsilon ) - R_{0}(v^{2} - i\varepsilon ))\).
 
2
Following [36], for any s 0C we use the notation
$$\displaystyle{\mbox{ Res}_{n}\Big\vert _{s=s_{0}}\zeta _{1}(s):= \left \{\begin{array}{c} \mbox{ coefficient of}\ (s - s_{0})^{-n} \\ \mbox{ in the Laurent expansion of}\ \zeta _{1}(s)\ \mbox{ at}\ s = s_{0} \end{array} \right \}\;.}$$
 
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Metadata
Title
Relative-Zeta and Casimir Energy for a Semitransparent Hyperplane Selecting Transverse Modes
Authors
Claudio Cacciapuoti
Davide Fermi
Andrea Posilicano
Copyright Year
2017
DOI
https://doi.org/10.1007/978-3-319-58904-6_5

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