Casimir self-energy of a δ−δ′ sphere

Vacuum energy of scalar fields on spherical shells with general matching conditions

“…In figure 2 we show a plot of the Casimir energy for R = 1 as a function of the parameter d when the parameter c satisfies the constraint found in (76). As it is clear from the graph, the energy is always positive as it has already been observed in [37].…”

“…This case coincides with the δ-δ ′ potential which has been considered in [37]. Here, we have to analyze the numerator in (66) in order to discern whether or not a range for the parameters (c, d, R) exists such that the residue vanishes.…”

“…By subtracting, and then adding, in the integrand of (36), the first N terms of the uniform asymptotic expansion of ln F ν (νz), derived in appendix B, one obtains an expression for the spectral zeta function in (33) which is valid in the semi-plane ℜ(s) > (D − N − 2)/2. Since the form of the uniform asymptotic expansion of ln F ν (νz) depends on whether b ̸ = 0 or b = 0 in (37) it is convenient to treat the two cases separately.…”

“…If a two-interval self-adjoint extension of (6) develops negative eigenvalues, they will appear as positive zeroes of the corresponding characteristic function F ν (νz) in (37). An attempt at determining whether positive zeroes exist for F ν (νz) in (37) for all allowed values of the parameters α, γ and b has proved to be surprisingly difficult. While we know that F ν (νz) tends to a constant value as z → 0, as shown in (35), and that…”

“…Nevertheless, one can qualitatively show that F ν (νz) is indeed a monotonic function that is increasing when b > 0 and decreasing when b < 0. This can be done by plotting the function (37) for a fixed positive value of b, and for several values of the remaining parameters, and by noticing that the resulting graphs are always monotone increasing. A similar approach is used for a fixed positive value of b where one observes that the ensuing graphs are always monotone decreasing.…”