The unified transform method for linear initial-boundary value problems: a spectral interpretation

TL;DR

This paper interprets the unified transform method for linear initial-boundary value problems as an extension of Fourier techniques, highlighting its spectral interpretation and applicability to non-self-adjoint operators.

Contribution

It provides a spectral interpretation of the unified transform method, extending classical Fourier analysis to non-self-adjoint operators in boundary value problems.

Findings

Unified transform method generalizes Fourier techniques for non-self-adjoint operators.

Spectral functionals enable diagonalization of certain non-self-adjoint differential operators.

The method applies to well-posed problems on finite intervals and half-lines.

Abstract

It is known that the unified transform method may be used to solve any well-posed initial-boundary value problem for a linear constant-coefficient evolution equation on the finite interval or the half-line. In contrast, classical methods such as Fourier series and transform techniques may only be used to solve certain problems. The solution representation obtained by such a classical method is known to be an expansion in the eigenfunctions or generalised eigenfunctions of the self-adjoint ordinary differential operator associated with the spatial part of the initial-boundary value problem. In this work, we emphasise that the unified transform method may be viewed as the natural extension of Fourier transform techniques for non-self-adjoint operators. Moreover, we investigate the spectral meaning of the transform pair used in the new method; we discuss the recent definition of a new class of spectral functionals and show how it permits the diagonalisation of certain non-self-adjoint spatial differential operators.