 Inverse spectral theory

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.