Spectral theory of some non-selfadjoint linear differential operators

TL;DR

This paper characterizes the spectral properties of non-selfadjoint linear differential operators with constant coefficients on bounded intervals, linking their spectral features to solutions of related boundary value problems and complex integral representations.

Contribution

It establishes a novel explicit correspondence between the spectral properties of such operators and the solutions of associated PDE boundary value problems, using a transform method.

Findings

Spectral properties are linked to the solution of PDE boundary value problems.

The eigenfunction family may form a basis depending on the boundary conditions.

A contour integral representation of solutions is used to analyze spectral properties.

Abstract

We give a characterisation of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary conditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator $S$ with the properties of the solution of a corresponding boundary value problem for the partial differential equation $\partial_t q \pm iSq=0$. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we consider its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas and one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator.