The spectral form of the functional model for maximally dissipative operators: A Lagrange identity approach

Abstract

The spectral and scattering properties of non-selfadjoint problems pose a mathematical challenge. Apart from exceptional cases, the well-developed methods used to examine the spectrum of selfadjoint problems are not applicable. One of the tools to attack non-selfadjoint problems are functional models. A drawback of many functional models is that their constructions require objects which may be difficult to describe explicitly, such as operator square roots, making it hard to apply the results to specific examples. We develop a functional model for the case when the non-selfadjointness arises both in additive terms and in the boundary conditions which is based on the Lagrange identity. The flexibility of the choice of the $\Gamma$-operators in the Lagrange identity means that these can be chosen so that expressions arising in the model are given explicitly in terms of physical parameters (coefficients, boundary conditions and Titchmarsh-Weyl $M$-function) of the maximally dissipative operator. The presentation of such explicit expressions for the spectral form of the functional model is arguably the main contribution of the present paper. In the spectral form of the functional model, the selfadjoint dilation is very simple, being the operator of multiplication by an independent variable in some auxiliary vector-valued function space. We also obtain an explicit expression for the completely non-selfadjoint part of the operator and an operator-analytic proof of the famous result by Sz.-Nagy-Foias on the pure absolute continuity of the spectrum of the minimal selfadjoint dilation. Finally, we consider an example of a limit circle Sturm-Liouville operator.