Pavel ExnerDoppler Institute for Mathematical Physics and Applied Mathematics, Prague

Geometric effects in soft waveguides spectra

Abstract:

Remembering one of the many topics Brian contributed to, I am going to discuss the discrete spectrum induced by geometric perturbations of systems that may regarded as a soft version of a Dirichlet quantum waveguides. I will focus on two particular examples. One is a two-dimensional Schr ̈odinger operator with the potential in the form of a channel of fixed profile built over a curve which is not straight but it is straight outside a compact. The second one concerns a sort of soft waveguide with a longitudinal periodic structure, specifically an array of potential wells. Here I again consider perturbations which preserve the array structure outside a compact region and in this case higher dimensions will be allowed. I will also mention some open problems.