Rectangular microwave resonators with magnetic anisotropy . Mapping onto pseudo integrable rhombus

The rectangular microwave resonator filed by a ferrite with uniaxial magnetic anisotropy is considered. It is shown that this task can be reduced to an empty rhombus resonator with vertex angle defined by external magnetic field provided the magnetic anisotropy of the ferrite is strong. Therefore statistics of eigen frequencies for TM modes is described by the Brody or semi-Poisson distribution with some exceptional cases. One of the main research lines in quantum chaos is to investigate the statistics of energy levels and eigen functions of quantum systems whose classical counterpart is chaotic. A very popular class of systems are two-dimensional Euclidean billiards, which is described by the Helmholtz equation −∇ψ(x, y) = λψ(x, y) (1) with Dirichlet boundary conditions ψ(x, y) = 0 for (x, y) at the boundary of billiard. It was shown that the eigen value statistics of λ obey the statistics of random matrix ensemble [1,2]. The distribution function for the amplitudes of the eigenfunctions ψ is perfectly well described by a Gaussian distribution [3,4]. Correspondingly the square ρ = |ψ|2 is describing by the well known Porter-Thomas (P-T) distribution [4] P (ρ) = 1 √ 2πρ exp(−ρ/2). (2) For integrable billiards, for example, rectangular one, the eigen value statistics is describing by the Poisson distribution. The distribution function of ρ is given by formula 1 g(ρ) = 1 π2 √ ρ K(1− ρ), 0 < ρ < 1 (3) E-mail address: [email protected], [email protected] 1A.I.Saichev, private communication