= 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> $\operatorname{diag}(1,3)$ and $\begin{pmatrix}2&1\\1&2\end{pmatrix}$ 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>.
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