Trans. Amer. Math. Soc. 365, 3923-3942 (2013) [DOI: 10.1090/S0002-9947-2012-05821-1]
Uniqueness Results for One-Dimensional Schrödinger Operators with Purely Discrete Spectra
We provide an abstract framework for singular one-dimensional Schrödinger operators with purely discrete spectra to show when the spectrum plus norming constants determine such an operator completely. As an example we apply our findings to prove new uniqueness results for perturbed quantum mechanical harmonic oscillators. In addition, we also show how to establish a Hochstadt-Lieberman type result for these operators. Our approach is based on the singular Weyl-Titchmarsh-Kodaira theory which is extended to cover the present situation.