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

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 $L^2(\mathbf{R}^d)$ ($d \geq 2$) 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 $α=α(-Δ_{\parallel})$, where $-Δ_{\parallel}$ is the free Laplacian in $L^2(\mathbf{R}^{d-1})$. Thus $α$ may depend on the components of the wave vector parallel to 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 $α=χ_{I}(-Δ_{\parallel})$, where $χ_{I}$ is the characteristic function of an interval $I$, the thermal Casimir energy can be explicitly computed.