Resonance photon generation in a vibrating cavity

The problem of photon creation from vacuum due to the non-stationary Casimir effect in an ideal one-dimensional Fabry-Perot cavity with vibrating walls is solved in the resonance case, when the frequency of vibrations is close to the frequency of some unperturbed electromagnetic mode: omega(w) = p(pi c/L-0)(1 + delta), \delta\ << 1, p = 1, 2, ... (L-0 is the mean distance between the walls). An explicit analytical expression for the total energy in all the modes shows an exponential growth if \delta\ is less than the dimensionless amplitude of vibrations epsilon << 1, the increment being proportional to p root epsilon(2)-delta(2). The rate of photon generation from vacuum in the (j + ps)th mode goes asymptotically to a constant value cp(2) sin(2) (pi j/p)root epsilon(2) - delta(2)/[pi L-0(j + ps)], the numbers of photons in the modes with indices p, 2p, 3p .... being the integrals of motion. The total number of photons in all the modes is proportional to p(3)(epsilon(2) - delta(2))t(2) in the short-time and in the long-time limits. In the case of strong detuning \delta\ > epsilon the total energy and the total number of photons generated from vacuum oscillate with the amplitudes decreasing as (epsilon/delta)(2) for epsilon << \delta\. The special cases of p = 1 and p = 2 are studied in detail.