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Gerald Teschl is a mathematician working on spectral theory and inverse spectral theory for differential and difference operators, together with their connections to integrable systems. His books treat Schrödinger operators, ordinary differential equations and discrete models such as the Toda lattice.^[ref] A direct spectral problem asks for the spectrum of an operator specified by its coefficients. An inverse spectral problem asks how the coefficients can be reconstructed from spectral data. This turns observed frequencies or quantum energy data into information about the underlying mathematical model, subject to the uniqueness conditions of the particular problem.

Inverse spectral theory reconstructs an operator or its coefficients from spectral data such as eigenvalues, spectral weights or boundary responses. It asks what the data determine and what additional information is needed for uniqueness.

Eigenvalues alone generally lose information. The matrices $\mathrm{diag}(1,3)$ and $\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)$ have the same eigenvalues but different entries in the fixed coordinate system. Orthogonal conjugation changes eigenvectors while preserving the spectrum. An inverse problem must therefore specify whether recovery is sought in fixed coordinates, up to a change of basis, or within a more restricted class. Spectral weights can retain information about how selected measurements couple to eigenvectors.