Uniqueness results for one-dimensional Schrödinger operators with purely discrete spectra (Q2846989)

Using the singular Weyl-Titchmarsh-Kodaira theory, the authors study the inverse spectral problem for 1-D Schrödinger operators having purely discrete spectrum. More precisely, consider the Schrödinger operator NEWLINE\[NEWLINE H=-\frac{d^2}{dx^2}+q(x),\quad x\in(a,b), NEWLINE\]NEWLINE on the Hilbert space \(L^2(a,b)\) with a real-valued potential \(q\in L^1_{\text{loc}}(a,b)\). Assume that \(H\) has purely discrete spectrum. If at least one of the endpoints is regular, then the classical Borg-Marchenko result states that the set of eigenvalues together with the set of corresponding norming constants uniquely determines the potential \(q\). The main result of the paper under review is the local Borg-Marchenko theorem in the case when both endpoints \(a\) and \(b\) are singular. This result enables the authors to prove new uniqueness results for perturbed quantum mechanical harmonic oscillators and perturbed Pöschel-Teller operators.