Advances in Quantum Mechanics pp 71-97 | Cite as
Relative-Zeta and Casimir Energy for a Semitransparent Hyperplane Selecting Transverse Modes
Claudio Cacciapuoti
Davide Fermi
Andrea Posilicano
We study the relative zeta function for the couple of operators A0 and A α , where A0 is the free unconstrained Laplacian in L2(R d ) (d ≥ 2) and A α is the singular perturbation of A0 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 L2(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.
Casimir effect Delta-interactions Finite temperature quantum fields Relative zeta function Zeta regularization
MSC 2010
81Q10 81T55 81T10
C.C. and D.F. acknowledge the support of the FIR 2013 project "Condensed Matter in Mathematical Physics", Ministry of University and Research of Italian Republic (code RBFR13WAET).
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 r1(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 e1(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
$$\displaystyle{ e_{1}(v) = -{ v \over 2\pi ^{2}}\;\arctan \left ({\tilde{\alpha }_{0} \over v}\right )\,\chi _{(0,+\infty )}(v)\;, }$$
where \(\tilde{\alpha }_{0}\) is defined according to Eq. (49). This shows that the map v ↦ e1(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 ζ1(s) in terms of e1(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 e1(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 ↦ ζ1(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 ζ1(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 \(\mathcal{E}(\beta )\).
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1.DiSAT, Sezione di MatematicaUniversità dell'InsubriaComoItaly
2.Dipartimento di MatematicaUniversità di MilanoMilanoItaly
Cacciapuoti C., Fermi D., Posilicano A. (2017) Relative-Zeta and Casimir Energy for a Semitransparent Hyperplane Selecting Transverse Modes. In: Michelangeli A., Dell'Antonio G. (eds) Advances in Quantum Mechanics. Springer INdAM Series, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-319-58904-6_5