Alexander Pushnitski (KCL)

The spectrum of the scattering matrix of the Schrodinger operator near resonant energies

Abstract:
The object of investigation is the spectrum of the scattering matrix S(E) of the Schrodinger operator with the potential satisfying assumptions typical in the study of shape resonances. I will discuss the spectrum of S(E) in the semiclassical limit when the energy E varies near some non-degenerate trapping energy value E_0. The main result is that as E increases and passes through E_0, exactly one eigenvalue of S(E) experiences an exponentially fast (in the semiclassical limit) anti-clockwise rotation by 2*\pi, while the other eigenvalues remain approximately constant. Due to the presence of avoided crossings, the result is formulated in terms of an eigenvalue counting function of S(E). The talk is based on a recent joint work with Shu Nakamura (University of Tokyo)