Schrodinger operator with complex potential and inner functions

The Schrodinger operator on the semiaxis with summable imaginary part of the potential admits a local trace scattering theory owing to the existence of boundary values for the corresponding analytic functions on the real axis (Sakhnovich, Pavlov, Naboko; late 60s – 70s). It turns out that if the imaginary part of the potential is not summable and does not change sign the characteristic function of the operator is purely inner. The methods used in the proof of this result, such as the perturbation theory for scalar multiples of matrix analytic functions and the spreading of the potential, are sketched in the talk. It is also supposed to discuss the problem of constructing the spectral subspaces for the operators of the type under consideration.