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 _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.

Appendix: The Case α = const. in Spatial Dimensions d = 3

In order to make connection with the existing literature, in the present appendix we briefly review the computation of the Casimir energy for the 3-dimensional model described in Eq. (51), that is

$$\displaystyle{\alpha (\rho ) =\alpha _{0}\quad \mbox{ for all}\ \rho \in [0,+\infty )\ \mbox{ and some}\ \alpha _{0}> 0\qquad (d = 3)\;.}$$

As a matter of fact, it can be easily checked that all the arguments described in the present manuscript continue to make sense also in this particular case, even though α does not fulfil the required assumptions since it does not have compact support.

First of all, let us notice that the integral representation (20) for the function r ₁(z) continues to make sense for any z ∈ C ∖[0, +∞) (the integral in the cited equation is trivially seen to remain finite in the present case). Then, by the same arguments of Sect. 3.1 one obtains an expression like Eq. (26) for the relative spectral measure e ₁(v); moreover, the term I _α appearing therein (given by Eq. (27) ) can be evaluated explicitly. This allows to infer for the relative spectral measure the expression

$$\displaystyle{ e_{1}(v) = -{ v \over 2\pi ^{2}}\;\arctan \left ({\tilde{\alpha }_{0} \over v}\right )\,\chi _{(0,+\infty )}(v)\;, }$$

(53)

where $\tilde{\alpha }_{0}$ is defined according to Eq. (49). This shows that the map v ↦ e ₁(v) is continuous on (0, +∞) and fulfils, for any N ∈ {0, 1, 2, …},

$$\displaystyle{ e_{1}(v) = \left \{\begin{array}{ll} O(v) &\quad \mbox{ for}\ v \rightarrow 0^{+}\;, \\ \sum _{n=0}^{N}{(-1)^{n+1}\,\tilde{\alpha }_{ 0}^{2n+1} \over 2\pi ^{2}(2n + 1)} \;v^{-2n} + O(v^{-2(N+1)})&\quad \mbox{ for}\ v \rightarrow +\infty \;. \\ \end{array} \right. }$$

Next, let us consider the representation (16) of the relative zeta function ζ ₁(s) in terms of e ₁(v); in view of the above considerations, it appears that the integral in the cited equation is finite for any complex s inside the strip

$$\displaystyle{ \Big\{s \in \mathbf{C}\;\Big\vert \;{1 \over 2} <\mathop{ \mathrm{Re}}\nolimits \,s <1\Big\}\;. }$$

To proceed, notice that for any such s the integral in Eq. (16) can be evaluated explicitly using the expression (53) for e ₁(v); one obtains

$$\displaystyle{ \zeta _{1}(s) = -{ \tilde{\alpha }_{0}^{2-2s} \over 8\pi \,(s - 1)\,\cos (\pi s)}\;, }$$

which determines the analytic continuation of s ↦ ζ ₁(s) to a function which is meromorphic on the whole complex plane, with simple pole singularities at s = 1 and s = ±1∕2, ±3∕2, … .

Using the above expression for ζ ₁(s) and Eqs. (44)–(45) for the thermal Casimir energy $\mathcal{E}(\beta )$, one easily infers the final result (52), that is

$$\displaystyle{\begin{array}{c} \mathcal{E}(\beta ) ={ \tilde{\alpha }_{0}^{3} \over 12\pi ^{2}}\left ({4 \over 3} -\log (2\ell\tilde{\alpha }_{0})\!\right ) -{ 1 \over 2\pi ^{2}}\int _{0}^{+\infty }dv\;{ v^{2} \over e^{\beta v} - 1}\;\arctan \left ({\tilde{\alpha }_{0} \over v}\right )\,. \\ \end{array} }$$

For completeness, let us mention that the above expression can be easily employed to derive the zero temperature expansion (β → +∞) of the Casimir energy $\mathcal{E}(\beta )$.