Scattering theory for a class of non-selfadjoint extensions of symmetric operators

TL;DR

This paper develops a functional model for certain non-selfadjoint extensions of symmetric operators, providing explicit formulas for wave scattering analysis, including wave operators and scattering matrices, applicable to both self-adjoint and non-self-adjoint cases.

Contribution

It introduces explicit formulas for the unitary group actions and constructs wave operators for a class of non-selfadjoint operator extensions, advancing scattering theory methods.

Findings

Derived explicit formulas for unitary group actions

Constructed wave operators for non-selfadjoint extensions

Presented a new representation for the scattering matrix

Abstract

This work deals with the functional model for a class of extensions of symmetric operators and its applications to the theory of wave scattering. In terms of Boris Pavlov's spectral form of this model, we find explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices. On the basis of these formulae, we are able to construct wave operators and derive a new representation for the scattering matrix for pairs of such extensions in both self-adjoint and non-self-adjoint situations.