Borg–Marchenko uniqueness theorem

Remarks

The function $m$ determines the spectral measure through its Herglotz representation, and the spectral measure determines $q$ by Gelfand–Levitan theory; this gives one proof when $q$ is regular enough. Example: if $m(z) = i\sqrt{z}$ for all non-real $z$, then $q = 0$ almost everywhere, since this is the $m$-function of $q = 0$. The hypothesis cannot be weakened to equality of the spectra: in general the spectrum alone does not determine $q$. There is also a local version: if $m_1(z) - m_2(z)$ decays fast enough as $z \to \infty$ along suitable rays, then $q_1 = q_2$ on an initial interval $[0, a]$.

Going deeper

For a student

This is the basic uniqueness result of inverse spectral theory for one-dimensional Schrödinger operators. It sits beside the Gelfand–Levitan reconstruction, which turns the same data into an explicit formula for $q$. The local version shows that the asymptotic behaviour of $m(z)$ for large $\abs{z}$ already determines $q$ on an initial interval.