It seems to be true, that if some eigenvalue of a Hamilton's operator is an isolated eigenvalue (part of discrete spectrum, not of continuous spectrum), then the corresponding eigenstate is normalizable, and on the other hand, if some eigenvalue of a Hamilton's operator is not isolated, then the corresponding eigenstate is not normalizable (not a vector of a Hilbert space) (like plane waves).(adsbygoogle = window.adsbygoogle || []).push({});

This starts to become intuitively clear once one has seen sufficiently examples obeying this pattern, but I have never encountered any general proof for this. Anyone knowing something about the proof?

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# Connection between isolated eigenvalues and normalizable eigenstates

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