The asymptotic behaviour of the scattering matrix in a neighbourhood of the endpoints of a spectral gap
The behaviour of the scattering matrix is investigated as the spectral parameter approaches an endpoint of a spectral gap of a quantum waveguide from the inside or the outside. The waveguide has two sleeves, one is cylindrical and the other periodic. When the spectral parameter traverses the spectral gap, the scattering matrix is reshaped because the number of waves inside and outside the gap is different. Notwithstanding, the smaller scattering matrix (in size) is transformed continuously into an identical block in the bigger scattering matrix and, in addition, the latter takes block diagonal form in the limit at the endpoint of the gap, that is, at the spectral threshold. The unexpected phenomena are related to the other block. It is shown that in the limit this block can only take certain values at the threshold, and taking one or other of these values depends on the structure of the continuous spectrum and also on the structure of the subspace of 'almost standing' waves at the threshold, which are solutions of the homogeneous problem that transfer no energy to infinity. A criterion for the existence of such solutions links the dimension of this subspace to the multiplicity of the eigenvalue −1 of the threshold scattering matrix. Asymptotic formulae are obtained, which show, in particular, that the phenomenon of anomalous scattering of high-amplitude waves at near-threshold frequencies, discovered by Weinstein in a special acoustic problem, also occurs in periodic waveguides. Bibliography: 38 titles. (paper)