Relative-Zeta and Casimir energy for a semitransparent hyperplane selecting transverse modes

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 $L^2({\mathbf{R}}^d)$ ($d\ge 2$) 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 $L^2({\mathbf{R}}^{d-1})$. Thus $\alpha$ may depend on the components of the wave vector parallel to 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.

21 pages

References in corpus (2)

• Local and Global Casimir Energies for a Semitransparent Cylindrical Shell
• Local zeta regularization and the scalar Casimir effect II. Some explicitly solvable cases

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• An introduction to the two-dimensional Schrödinger equation with nonlinear point interactions