F. Haake et al., MICROWAVE BILLIARDS WITH BROKEN TIME-REVERSAL INVARIANCE, Journal of physics. A, mathematical and general, 29(18), 1996, pp. 5745-5757
We consider a microwave resonator with three single-channel waveguides
attached. One of these serves to couple waves into and out of the res
onator; the remaining two are connected to form a one-way handle so as
to break time reversal invariance. The poles of the input-output scat
tering coefficient of such a resonator are shown to be the eigenvalues
of a non-Hermitian effective 'Hamiltonian' H-eff = H - i Gamma, the a
nti-Hermitian part Gamma of which has rank 1 and is responsible for th
e breaking of time reversal invariance. All of the spectral statistics
recently observed for such a microwave billiard are reproduced quanti
tatively by taking H and Gamma as random matrices. In particular, the
distribution of nearest-neighbour spacings of the resonances is close
to that of the GUE when H belongs to the GOE corresponding to a Sinai
shape of the resonator; linear level repulsion results when H belongs
to the Poissonian ensemble as it corresponds to a rectangular resonato
r.