We study the relative zeta function for the couple of operators $A_0$ and $A_{\alpha}$, where $A_0$ is the free unconstrained Laplacian in $L^2(R^d)$ ($d >= 2$) and $A_{\alpha}$ 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 \alpha, which is related to the strength of the perturbation, is of the kind $\alpha = \alpha(-Delta_{\parallel})$, where $\Delta_{\parallel}$ is the free Laplacian in $L^2(R^{d-1})$. Thus $\alpha$ may depend on the components of the wave vector parallel to the hyperplane; in this sense $A_{\alpha}$ describes a semitransparent hyperplane selecting transverse modes. As an application we give an expression for the associated thermal Casimir energy. Whenever $\alpha = \chi_{I}(-\Delta_{\parallel})$, where $\chi_{I}$ is the characteristic function of an interval $I$, the thermal Casimir energy can be explicitly computed.

Relative-zeta and casimir energy for a semitransparent hyperplane selecting transverse modes / C. Claudio, D. Fermi, P. Andrea (SPRINGER INDAM SERIES). - In: Advances in Quantum Mechanics : Contemporary Trends and Open Problems / [a cura di] A. Michelangeli, G. Dell'Antonio. - Prima edizione. - [s.l] : Springer International Publishing, 2017. - ISBN 9783319589039. - pp. 71-97 [10.1007/978-3-319-58904-6_5]

Relative-zeta and casimir energy for a semitransparent hyperplane selecting transverse modes

D. Fermi
;
2017

Abstract

We study the relative zeta function for the couple of operators $A_0$ and $A_{\alpha}$, where $A_0$ is the free unconstrained Laplacian in $L^2(R^d)$ ($d >= 2$) and $A_{\alpha}$ 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 \alpha, which is related to the strength of the perturbation, is of the kind $\alpha = \alpha(-Delta_{\parallel})$, where $\Delta_{\parallel}$ is the free Laplacian in $L^2(R^{d-1})$. Thus $\alpha$ may depend on the components of the wave vector parallel to the hyperplane; in this sense $A_{\alpha}$ describes a semitransparent hyperplane selecting transverse modes. As an application we give an expression for the associated thermal Casimir energy. Whenever $\alpha = \chi_{I}(-\Delta_{\parallel})$, where $\chi_{I}$ is the characteristic function of an interval $I$, the thermal Casimir energy can be explicitly computed.
Casimir effect; Delta-interactions; Finite temperature quantum fields; Relative zeta function; Zeta regularization; Mathematics (all)
Settore MAT/07 - Fisica Matematica
Settore MAT/05 - Analisi Matematica
Settore FIS/02 - Fisica Teorica, Modelli e Metodi Matematici
2017
Book Part (author)
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/541608
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