In this work we consider the dynamical Casimir effect for a massless scalar field -- under Dirichlet boundary conditions -- between two concentric spherical shells. We obtain a general expression for the average number of particle creation, for an arbitrary law of radial motion of the spherical shells, using two distinct methods: by computing the density operator of the system and by calculating the Bogoliubov coefficients. We apply our general expression to breathing modes: when only one of the shells oscillates and when both shells oscillate in or out of phase. We also analyze the number of particle production and compare it with the results for the case of plane geometry.
We study the finite temperature Casimir interaction between two concentric cylinders. When the separation between the cylinders is much smaller than the radii of the cylinders, the asymptotic expansions of the Casimir interaction are derived. Both the low temperature and the high temperature regions are considered. The leading terms are found to agree with the proximity force approximations. The low temperature leading term of the temperature correction is also computed and it is found to be independent of the boundary conditions imposed on the larger cylinder.
We analyze the dynamical Casimir effect for a massless scalar field confined between two concentric spherical shells which impose on the field mixed boundary conditions. We thus complement a previous result [Phys. Rev. A \textbf{78}, 032521 (2008)], where the same problem was considered but in that case the field was submitted to a Dirichlet boundary condition in both moving spherical shells. A general expression for the average number of created particles is deduced for an arbitrary law of radial motion of the spherical shells. This expression is then applied to harmonic oscillations of the s…
The Casimir interaction between two perfectly conducting, infinite, concentric cylinders is computed using a semiclassical approximation that takes into account families of classical periodic orbits that reflect off both cylinders. It is then compared with the exact result obtained by the mode-by-mode summation technique. We analyze the validity of the semiclassical approximation and show that it improves the results obtained through the proximity theorem.
Motivated by the problem of Casimir energy, we investigate the idea of using inhomogeneity of surfaces instead of their corrugation, which leads to Casimir interaction between two inhomogeneous semi-transparent concentric cylinders. Using the multiple scattering method, we study the Casimir energy and torque between the cylinders with different potentials subjected to Dirichlet boundary conditions, both in weak and strong coupling regimes. We also extend our formalism to the case of two inhomogeneous dielectrics in a weak coupling regime.
The Casimir effect giving rise to an attractive force between the closely spaced two concentric spheres that confine the massless scalar field is calculated by using a direct mode summation with contour integration in the complex plane of eigenfrequencies. We devoleped a new approach appropriate for the calculation of the Casimir energy for spherical boundary conditions. The Casimir energy for a massless scalar field between the closely spaced two concentric spheres coincides with the Casimir energy of the parallel plates for a massless scalar field in the limit when the dimensionless paramete…
The Casimir interaction energy due to the vacuum fluctuations of a massive vector field between two perfectly conducting concentric spherical bodies is computed. The TE contribution to the Casimir interaction energy is a direct generalization of the massless case but the TM contribution is much more complicated. Each TM mode is a linear combination of a transverse mode which is the generalization of a TM mode in the massless case and a longitudinal mode that does not appear in the massless case. In contrast to the case of two parallel perfectly conducting plates, the are no TM discrete modes t…
The Casimir energy for a massless scalar field between the closely spaced two concentric D-dimensional (for D>3) spheres is calculated by using the mode summation with contour integration in the complex plane of eigenfrequencies and the generalized Abel-Plana formula for evenly spaced eigenfrequency at large argument. The sign of the Casimir energy between closely spaced two concentric D-dimensional spheres for a massless scalar field satisfying the Dirichlet boundary conditions is strictly negative. The Casimir energy between D-1 dimensional surfaces close to each other is regarded as interes…
We consider the finite temperature Casimir effect between two concentric spheres due to the vacuum fluctuations of the electromagnetic field in the $(D+1)$-dimensional Minkowski spacetime. Different combinations of perfectly conducting and infinitely permeable boundary conditions are imposed on the spheres. The asymptotic expansions of the Casimir free energies when the dimensionless parameter $\vep$, the ratio of the distance between the spheres to the radius of the smaller sphere, is small are derived in both the high temperature region and the low temperature region. It is shown that the le…
We study the Casimir effect at finite temperature for a massless scalar field in the parallel plates geometry in N spatial dimensions, under various combinations of Dirichlet and Neumann boundary conditions on the plates. We show that in all these cases the entropy, in the limit where energy equipartitioning applies, is a geometrical factor whose sign determines the sign of the Casimir force.
We utilize Effective Field Theory(EFT) techniques to calculate the casimir torque on a cylindrical gear in the presence of a polarizable but neutral object. We present results for the energy and torque as a function of angle for a gear with multiple cogs, as well as for the case of a concentric cylindrical gear.
We compute the Casimir interaction energy between two perfectly conducting, concentric cylinders, using the mode-by-mode summation technique. Then we compare it with the approximate results obtained using the proximity theorem and a semiclassical approximation based on classical periodic orbits. We show that the proximity theorem with a particular choice for the effective area coincides with the semiclassical approximation and reproduces the exact result far beyond its expected range of validity. We also compute the force between slightly eccentric cylinders and discuss the advantages of using…
We numerically evaluate the Casimir interaction energy for configurations involving two perfectly conducting eccentric cylinders and a cylinder in front of a plane. We consider in detail several special cases. For quasi-concentric cylinders, we analyze the convergence of a perturbative evaluation based on sparse matrices. For concentric cylinders, we obtain analytically the corrections to the proximity force approximation up to second order, and we present an improved numerical procedure to evaluate the interaction energy at very small distances. Finally, we consider the configuration of a cyl…
In this paper we investigate the vacuum polarization effects associated with a massive fermionic field due to the non-trivial topology of the global monopole spacetime and boundary conditions imposed on this field. Specifically we investigate the vacuum expectation values of the energy-momentum tensor and fermionic condensate admitting that the field obeys the MIT bag boundary condition on two concentric spherical shells. In order to develop this analysis, we use the generalized Abel-Plana summation, which allows to extract from the vacuum expectation values the contribution coming from a sing…
The effect of slope for the rack gear in the massless scalar field model is considered. It appears, that the slope of profile surfaces can essentially change the value of normal Casimir force, whereas average value of tangential force remains almost unchanged. At the same time we observe essential shift of maximum and minimum tangential force positions.
We calculate the Casimir interaction between two short range scatterers embedded in a background of one dimensional massless Dirac fermions using a force operator approach. We obtain the force between two finite width square barriers, and take the limit of zero width and infinite potential strength to study the Casimir force mediated by the fermions. For the case of identical scatterers we recover the conventional attractive one dimensional Casimir force. For the general problem with inequivalent scatterers we find that the magnitude and sign of this force depend on the relative spinor polariz…
The quantum vacuum effects are investigated for a massive scalar field with general curvature coupling and obeying the Robin boundary conditions given on two concentric spherical shells with radii $a $ and $b$ in the $D+1$-dimensional global monopole background. The expressions are derived for the Wightman function, the vacuum expectation values of the field square, the vacuum energy density, radial and azimuthal stress components in the region between the shells. A regularization procedure is carried out by making use of the generalized Abel-Plana formula for the series over zeros of combinat…
In this paper, we consider the fermionic Casimir effect under a new type of space-time topology using the concept of quotient topology. The relation between the new topology and that in Ref. \cite{Feng,Zhai3} is something like that between a Möbius strip and a cylindric. We obtain the exact results of the Casimir energy and force for the massless and massive Dirac fields in the ($D+1$)-dimensional space-time. For both massless and massive cases, there is a $Z_2$ symmetry for the Casimir energy. To see the effect of the mass, we compare the result with that of the massless one and we found that…
We study the Casimir piston for massless scalar fields obeying Dirichlet boundary conditions in a three dimensional cavity with sides of arbitrary lengths $a,b$ and $c$ where $a$ is the plate separation. We obtain an exact expression for the Casimir force on the piston valid for any values of the three lengths. As in the electromagnetic case with perfect conductor conditions, we find that the Casimir force is negative (attractive) regardless of the values of $a$, $b$ and $c$. Though cases exist where the interior contributes a positive (repulsive) Casimir force, the total Casimir force on the …
We review recent work on the Casimir interaction energy between cylindrical shells. We include proposals for future experiments involving cylinders, such us a null experiment using quasi-concentric cylinders, a cylinder in front a conducting plate, and a cylindrical version of the rack and pinion powered by Casimir lateral force. We also present an exact formula for the theoretical evaluation of the vacuum interaction energy between eccentric cylindrical shells, and describe improved analytical and numerical evaluations for the particular case of concentric cylinders.
We derive a simplified general expression for the two-body scalar Casimir energy in generalized separable coordinate systems. We apply this technique to the case of radial semi-transparent planes in the annular region between two concentric Dirichlet cylinders. This situation is explored both analytically and numerically.
The Casimir effect for a massless scalar field on the helix boundary condition which is named as quantum spring is studied in our recent paper\cite{Feng}. In this paper, the Casimir effect of the quantum spring is investigated in $(D+1)$-dimensional spacetime for the massless and massive scalar fields by using the zeta function techniques. We obtain the exact results of the Casimir energy and Casimir force for any $D$, which indicate a $Z_2$ symmetry of the two space dimensions. The Casimir energy and Casimir force have different expressions for odd and even dimensional space in the massless c…
In the present paper, we show that a partially reflecting static mirror with time-dependent properties can produce, via dynamical Casimir effect in the context of a massless scalar field in $1+1$ dimensions, a larger number of particles than a perfectly reflecting one. As particular limits, our results recover those found in the literature for a perfect static mirror imposing a generalized or an usual time-dependent Robin boundary condition.
We investigate the thermal Casimir effect of a massless scalar field for two parallel plates moving in the equatorial orbit in Kerr space-time. Under the assumption that the typical cavity size is much smaller than the orbital radius, proposed by Sorge, we deduce the analytical expression of the renormalized free energy in this curved space-time. We also get the analytical representation for the renormalized internal energy, and find that there is a thermal correction to the Casimir energy, which depends on the proper temperature and the proper geometrical parameters of the plates. The asympto…
Casimir forces of massive fermionic Dirac fields are calculated for parallel plates geometry in spatial space with dimension d and imposing bag model boundary conditions.It is shown that in the range of ma>>1 where m is mass of fields quanta and a is the separation distance of the plates,it is equal to massive bosonic fields Casimir force for each degree of freedom.We argue this equality exists for any massive anyonic field in two-dimensional spatial space.Also the ratio of massless fermionic field Casimir force to its bosonic correspondent in d-dimentional spatial space is $(1-1/2^d)$.
The dynamical Casimir effect for a massless scalar field in 1+1-dimensions is studied numerically by solving a system of coupled first-order differential equations. The number of scalar particles created from vacuum is given by the solutions to this system which can be found by means of standard numerics. The formalism already used in a former work is derived in detail and is applied to resonant as well as off-resonant cavity oscillations.
The phenomenon of particle creation within a resonantly vibrating lossy cavity is investigated for the example of a massless scalar field at finite temperature. Leakage is provided by insertion of a dispersive mirror into a larger ideal cavity. Via the rotating wave approximation we demonstrate that for the case of parametric resonance the exponential growth of the number of created particles and the strong enhancement at finite temperatures are preserved in the presence of reasonable losses. The relevance for experimental tests of quantum radiation via the dynamical Casimir effect is addresse…
We calculate the exact Casimir interaction energy between two perfectly conducting, very long, eccentric cylindrical shells using a mode summation technique. Several limiting cases of the exact formula for the Casimir energy corresponding to this configuration are studied both analytically and numerically. These include concentric cylinders, cylinder-plane, and eccentric cylinders, for small and large separations between the surfaces. For small separations we recover the proximity approximation, while for large separations we find a weak logarithmic decay of the Casimir interaction energy, typ…
We define irreducible N-body spectral functions and Casimir energies and consider a massless scalar quantum field interacting locally by positive potentials with classical objects. Irreducible N-body spectral functions in this case are shown to be conditional probabilities of random walks. The corresponding irreducible contributions to scalar many-body Casimir energies are finite and positive/negative for an odd/even number of objects. The force between any two finite objects separable by a plane is always attractive in this case. Analytical and numerical world-line results for the irreducible…
The Casimir free energy for a system of two dielectric concentric nonmagnetic spherical bodies is calculated with use of a quantum statistical mechanical method, at arbitrary temperature. By means of this rather novel method, which turns out to be quite powerful (we have shown this to be true in other situations also), we consider first an explicit evaluation of the free energy for the static case, corresponding to zero Matsubara frequency ($n=0$). Thereafter, the time-dependent case is examined. For comparison we consider the calculation of the free energy with use of the more commonly known …
The Casimir force between two perfectly reflecting parallel plates is considered. In a recent paper we presented generalised physical boundary conditions describing perfectly reflecting parallel plates. These boundary conditions are applicable to a field possessing any spin, and include the well-known spin-1/2 and spin-1 boundary conditions as special cases. Here we use these general boundary conditions to show that the allowed values of energy-momentum turn out to be the same for any massless fermionic field and the same for any massless bosonic field. As a result one expects to obtain only t…
We study the Casimir effect in the presence of an extra dimension compactified on a circle of radius R ($M^4\times S^1$ spacetime). Our starting point is the Kaluza Klein decomposition of the 5D Maxwell action into a massless sector containing the 4D Maxwell action and an extra massless scalar field and a Proca sector containing 4D gauge fields with masses $m_n=n/R$ where $n$ is a positive integer. An important point is that, in the presence of perfectly conducting parallel plates, the three degrees of freedom do not yield three discrete (non-penetrating) modes but two discrete modes and one c…
Since massive scalar field plays an important role in theoretical physics, we consider the interaction between a sphere and a plate due to the vacuum fluctuation of a massive scalar field. We consider combinations of Dirichlet and Neumann boundary conditions. There is a simple prescription to obtain the functional formulas for the Casimir interaction energies, known as TGTG formula, for the massive interactions from the massless interactions. From the TGTG formulas, we discuss how to compute the small separation asymptotic expansions of the Casimir interaction energies up to the next-to-leadin…
The leading semiclassical estimates of the electromagnetic Casimir stresses on a spherical and a cylindrical metallic shell are within 1% of the field theoretical values. The electromagnetic Casimir energy for both geometries is given by two decoupled massless scalars that satisfy conformally covariant boundary conditions. Surface contributions vanish for smooth metallic boundaries and the finite electromagnetic Casimir energy in leading semiclassical approximation is due to quadratic fluctuations about periodic rays in the interior of the cavity only. Semiclassically the non-vanishing Casimir…
We consider a real massless scalar field in 1+1 dimensions satisfying time-dependent Robin boundary condition at a static wall. This condition can simulate moving reflecting mirrors whose motions are determined by the time-dependence of the Robin parameter. We show that particles can be created from vacuum, characterizing in this way a dynamical Casimir effect.
We study the zero point energy of massless scalar and vector fields subject to spheroidal boundary conditions. For massless scalar fields and small ellipticity the zero-point energy can be found using both zeta function and Green's function methods. The result agrees with the conjecture that the zero point energy for a boundary remains constant under small deformations of the boundary that preserve volume (the boundary deformation conjecture), formulated in the case of an elliptic-cylindrical boundary. In the case of massless vector fields, an exact solution is not possible. We show that a zon…
We study the Casimir effect due to a massive vector field in a system of two parallel plates made of real materials, in an arbitrary magnetodielectric background. The plane waves satisfying the Proca equations are classified into transverse modes and longitudinal modes which have different dispersion relations. Transverse modes are further divided into type I and type II corresponding to TE and TM modes in the massless case. For general magnetodielectric media, we argue that the correct boundary conditions are the continuities of $\mathbf{H}_{\parallel}, ϕ, \mathbf{A}$ and $\pa_xA_x$, where $x…
We consider the Casimir effect between two parallel plates localized on a brane. In order to properly compute the contribution to the Casimir energy due to any higher dimensional field, it is necessary to take into account the localization properties of the KK modes. When no massless mode appears in the spectrum, the correction to the Casimir energy is exponentially suppressed. When a massless mode is present in the spectrum, the correction to the Casimir energy can be, in principle, sizeable. Here we illustrate a new method to compute the correction to the Casimir energy between two parallel …
We consider a massless scalar field obeying Dirichlet boundary conditions on the walls of a two-dimensional L x b rectangular box, divided by a movable partition (piston) into two compartments of dimensions a x b and (L-a) x b. We compute the Casimir force on the piston in the limit L -> infinity. Regardless of the value of a/b, the piston is attracted to the nearest end of the box. Asymptotic expressions for the Casimir force on the piston are derived for a << b and a >> b.
The vacuum energy of a bosonic field interacting locally with objects is decomposed into irreducible $N$-body parts. The irreducible $N$-body contribution to the vacuum energy is finite if the common intersection $O_1\cap O_2...\cap O_N$ of all $N$ objects $O_i,i=1,..., N$ is empty. I prove that the perturbative expansion of the corresponding irreducible $N$-body spectral function $\tphi^{(N)}(β)$ for $β\sim 0$ vanishes to all orders even if some of the objects intersect. These irreducible spectral functions and their associated Casimir energies in principle can be computed numerically or appr…
Quantum vacuum energy has been known to have observable consequences since 1948 when Casimir calculated the force of attraction between parallel uncharged plates, a phenomenon confirmed experimentally with ever increasing precision. Casimir himself suggested that a similar attractive self-stress existed for a conducting spherical shell, but Boyer obtained a repulsive stress. Other geometries and higher dimensions have been considered over the years. Local effects, and divergences associated with surfaces and edges have been investigated by several authors. Quite recently, Graham et al. have re…
We consider a quantum massless fermionic field in (1+1) dimensions in the case of moving boundaries. We work in the canonical approach in order to find a Hamiltonian describing the dynamics of the field. Thus, we study the statistics of work and particles produced from the vacuum by a linear driving of a boundary.
The lowest radiative correction to the Casimir energy density between two parallel plates is calculated using effective field theory. Since the correlators of the electromagnetic field diverge near the plates, the regularized energy density is also divergent. However, the regularized integral of the energy density is finite and varies with the plate separation L as 1/L^7. This apparently paradoxical situation is analyzed in an equivalent, but more transparent theory of a massless scalar field in 1+1 dimensions confined to a line element of length L and satisfying Dirichlet boundary conditions.
We consider here, a two-atom system is uniformly moving through a circular ring at an ultra-relativistic speed and weakly interacting with common external fields. The vacuum fluctuations of the quantum fields generate the entanglement between the atoms. Hence an effective energy shift is originated, which depends on the inter-atomic distance. This is commonly known as resonance Casimir-Polder interaction (RCPI). It is well known that, for a linearly accelerated system coupled with a massless scalar field, we get a thermal response when the local inertial approximation is valid. On the contrary…
We present two novel models for repulsive Casimir interaction between positive perturbations. One example relies on non locality of the dielectric response and one relies on interference between (attractive) modes. Such examples are impossible to achieve in 1d massless theories, as they are prohibited by a generalization of the result of Lambrecht, Jaekel and Reynaud [Phys.Lett. A 225 (1997)] to non local dielectrics and to negative perturbations
We investigate, in the context of a real massless scalar field in $1+1$ dimensions, models of partially reflecting mirrors simulated by Dirac $δ-δ^{\prime}$ point interactions. In the literature, these models do not exhibit full transparency at high frequencies. In order to provide a more realistic feature for these models, we propose a modified $δ-δ^{\prime}$ point interaction that enables full transparency in the limit of high frequencies. Taking this modified $δ-δ^{\prime}$ model into account, we investigate the Casimir force, comparing our results with those found in the literature.
In this work, we consider the Casimir effect due to massless fermionic fields in the presence of long cylinders. More precisely, we consider the interaction between a cylinder parallel to a plate, between two parallel cylinders outside each other, and between a cylinder lying parallelly inside another cylinder. We derive the explicit formulas for the Casimir interaction energies and compute the leading and the next-to-leading order terms of the small separation asymptotic expansions. As expected, the leading order terms coincide with the proximity force approximations. We compare the results o…
The Casimir effect arises not only in the presence of material boundaries but also in space with nontrivial topology. In this paper, we choose a topology of the flat $(D+1)$-dimensional spacetime, which causes the helix boundary condition for a Hermitian massless scalar field. Especially, Casimir effect for a massless scalar field on the helix boundary condition is investigated in two and three dimensions by using the zeta function techniques. The Casimir force parallel to the axis of the helix behaves very much like the force on a spring that obeys the Hooke's law when the ratio $r$ of the pi…
Creation of scalar massless particles in two-dimensional Minkowski space-time--as predicted by the dynamical Casimir effect--is studied for the case of a semitransparent mirror initially at rest, then accelerating for some finite time, along a specified trajectory, and finally moving with constant velocity. When the reflection and transmission coefficients are those in the model proposed by Barton, Calogeracos, and Nicolaevici [$r(w)=-iα/(\w+iα)$ and $s(w)=\w/(\w+iα)$, with $α\geq 0$], the Bogoliubov coefficients on the back side of the mirror can be computed exactly. This allows us to prove t…
We study the Casimir energy of a massless scalar field that obeys Dirichlet boundary conditions on a hyperboloid facing a plate. We use the optical approximation including the first six reflections and compare the results with the predictions of the proximity force approximation and the semi-classical method. We also consider finite size effects by contrasting the infinite with a finite plate. We find sizable and qualitative differences between the new optical method and the more traditional approaches.
We compute the leading radiative correction to the Casimir force between two parallel plates in the $λΦ^4$ theory. Dirichlet and periodic boundary conditions are considered. A heuristic approach, in which the Casimir energy is computed as the sum of one-loop corrected zero-point energies, is shown to yield incorrect results, but we show how to amend it. The technique is then used in the case of periodic boundary conditions to construct a perturbative expansion which is free of infrared singularities in the massless limit. In this case we also compute the next-to-leading order radiative correct…
Motivated by experiments in which moving boundaries are simulated by time-dependent properties of static systems, we discuss the model of a massless scalar field submitted to a time-dependent Robin boundary condition (BC) at a static mirror in 1+1 dimensions. Using a perturbative approach, we compute the spectral distribution of the created particles and the total particle creation rate, considering a thermal state as the initial field state.
The Casimir energy for a massless, neutral scalar field in presence of a point interaction is analyzed using a general zeta-regularization approach developed in earlier works. In addition to a regular bulk contribution, there arises an anomalous boundary term which is infinite despite renormalization. The intrinsic nature of this anomaly is briefly discussed.
In this paper the quantum vacuum energies induced by massive fluctuations of one real scalar field on a configuration of two partially transparent plates are analysed. The physical properties of the infinitely thin plates are characterized by two Dirac-$δ$ potentials. We find that an attractive/repulsive Casimir force arises between the plates when the weights of the $δ$'s have equal/different sign. If some of the plates absorbs fluctuations below some threshold of energy (the corresponding weight is negative) there is the need to set a minimum mass to the scalar field fluctuations to preserve…
The Casimir mutual free energy F for a system of two dielectric concentric nonmagnetic spherical bodies is calculated, at arbitrary temperatures. Whereas F has recently been evaluated for the special case of metals (refractive index n=\infty), here analogous results are presented for dielectrics, and shown graphically when n=2.0. Our calculational method relies upon quantum statistical mechanics. The Debye expansions for the Riccati-Bessel functions when carried out to a high order are found to be very useful in practice (thereby overflow/underflow problems are easily avoided), and also to giv…
We derive the radiation pressure force on a non-relativistic moving plate in 1+1 dimensions. We assume that a massless scalar field satisfies either Dirichlet or Neumann boundary conditions (BC) at the instantaneous position of the plate. We show that when the state of the field is invariant under time translations, the results derived for Dirichlet and Neumann BC are equal. We discuss the force for a thermal field state as an example for this case. On the other hand, a coherent state introduces a phase reference, and the two types of BC lead to different results.
We show that even small nonlinearities significantly affect particle production in the dynamical Casimir effect at large evolution times. To that end, we derive the effective Hamiltonian and resum leading loop corrections to the particle flux in a massless scalar field theory with time-dependent Dirichlet boundary conditions and quartic self-interaction. To perform the resummation, we assume small deviations from the equilibrium and employ a kind of rotating wave approximation. Besides that, we consider a quantum circuit analog of the dynamical Casimir effect, which is also essentially nonline…
We investigate combined effects of nontrivial topology, induced by a cosmic string, and boundaries on the fermionic condensate and the vacuum expectation value (VEV) of the energy-momentum tensor for a massive fermionic field. As geometry of boundaries we consider two plates perpendicular to the string axis on which the field is constrained by the MIT bag boundary condition. By using the Abel-Plana type summation formula, the VEVs in the region between the plates are decomposed into the boundary-free and boundary-induced contributions for general case of the planar angle deficit. The boundary-…
In this work we study the Casimir effect for massless scalar fields propagating in a piston geometry of the type $I\times N$ where $I$ is an interval of the real line and $N$ is a smooth compact Riemannian manifold. Our analysis represents a generalization of previous results obtained for pistons configurations as we consider all possible boundary conditions that are allowed to be imposed on the scalar fields. We employ the spectral zeta function formalism in the framework of scattering theory in order to obtain an expression for the Casimir energy and the corresponding Casimir force on the pi…
