Casimir pistons with general boundary conditions

Abstract

In this work we analyze the Casimir energy and force for a scalar field endowed with general self-adjoint boundary conditions propagating in a higher dimensional piston configuration. The piston is constructed as a direct product I × N , with I = [0, L] ⊂ R and N a smooth, compact Riemannian manifold with or without boundary. The study of the Casimir energy and force for this configuration is performed by employing the spectral zeta function regularization technique. The obtained analytic results depend explicitly on the spectral zeta function associated with the manifold N and the parameters describing the general boundary conditions imposed. These results are then specialized to the case in which the manifold N is a d-dimensional sphere.

Hyperspherical δ-δ′ potentials

“…The theory of selfadjoint extensions for symmetric operators has been well known to mathematicians for many years. However, it only became a valuable tool for modern quantum physics after the seminal works of Asorey et al [15,17,18], in which the problem was re-formulated in terms of physically meaningful quantities for relevant operators in quantum mechanics and quantum field theory [26][27][28][29][30][31].…”

The Casimir effect for pistons with transmittal boundary conditions

“…It seems natural that the next step in this investigation would consist in considering more general boundary conditions at the endpoints of the piston configuration. Such generalized boundary conditions can be written as a linear combination, through real coefficients, of the values of the field and its derivative at the given endpoint (see for instance [21]). Unfortunately, finding a suitable range of values for the real coefficients characterizing the general boundary conditions and the parameter U that leads to a self-adjoint boundary value problem proves to be a prohibitive task within the formalism employed in this work.…”

“…By analyzing the behavior of z −2s ∂ z ln Ω ( j) λ (iλz, a) for z → 0 and z → ∞ it is not very difficult to show (see e.g. [21]) that the integral representation (3.2) is valid in the strip 1/2 < ℜ(s) < 1. In order to extend the region of convergence of the integral (3.2) to ℜ(s) ≤ 1/2, we subtract, and add, in the representation (3.2), a suitable number of terms of the asymptotic expansion of ln Ω…”

Gluing Formula for Casimir Energies

“…Many configurations, such as flat pistons at zero temperature [11][12][13][14][15][16][17][18][19][20] or finite temperatures [21][22][23][24], as well as curved pistons [25][26][27][28][29], have been analyzed on the basis of the spectrum of a Laplace-type operator associated with M 1 and M 2 . It is the aim of this article to introduce a completely new perspective on this type of analysis.…”