I was wondering if the good old quadratic potential was the only potential with equally spaced eigenvalues. Obviously you can construct others, such as a potential that is infinite in some places and quadratic in others, but that's only trivially different. I am not referring to equally spaced as a limiting behavior either, I mean truly integer spaced.
Any ideas? If not, is there a proof for its uniqueness?
If there are other potentials with equally spaced eigenvalues, can one use them as starting points for a free-field QFT? It would be interesting to know if there is a deeper mathematical relation between all of these potentials and whether they could be used to study interacting systems.
Answer
According to Inverse spectral theory as influenced by Barry Simon Fritz Gesztesy (Submitted on 2 Feb 2010) page 4:
A particularly interesting open problem in inverse spectral theory concerns the characterization of the isospectral class of potentials $V$ with purely discrete spectra (e.g., the harmonic oscillator $V(x) = x^2$ ).
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