Etudes for the inverse spectral problem

Abstract

In this note, we study inverse spectral problems for canonical Hamiltonian systems, which encompass a broad class of second-order differential equations on a half-line. Our goal is to extend the classical results developed in the work of Marchenko, Gelfand–Levitan, and Krein to broader classes of canonical systems and to illustrate the solution algorithms and formulae with a variety of examples. One of the main ingredients of our approach is the use of truncated Toeplitz operators, which complement the standard toolbox of the Krein–de Branges theory of canonical systems.

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