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In general spectral analysis, we have examples of unbounded from below hamiltonians with discrete spectrum. Is it okay to say that they have no sense in physical context, because for me it looks like particle can emmit infinite energy in some weird sense.

Qmechanic
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Георгий Леонидович Стависский
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    The Dirac coloumb Hamiltonian is unbounded from below but it gives good values for the spectrum of atoms – amilton moreira Nov 23 '23 at 05:52
  • Cool! Will learn more. As i understand, hamiltonian is unbounded, but spectrum is? Because unbounded spectra for physics is still not in my field of understanding) – Георгий Леонидович Стависский Nov 23 '23 at 05:59
  • The spectrum is unbounded below. Negative energy eingenstates is interpreted as anti particles – amilton moreira Nov 23 '23 at 06:40
  • See this for example and links therein. The Dirac Hamiltonian is bounded from below @amiltonmoreira...(at least if we talk about QFT and possibly after normal ordering) – Tobias Fünke Nov 23 '23 at 08:09
  • I saw information that this Hamiltonian is unbounded from below... See this source: https://chempedia.info/info/dirac_coulomb/. I saw your example and kind-of agree with it, but speaking formally, unbounded from below Hamiltonians still can posess adequate spectrum. But often, they dont, as a simple example: p^2/2m + x^2 - x^3 - a naive perturbation of oscillator leads to a hamiltonian without discrete spectrum at all, and it can be easily proved with insertion of trial gaussian functions with variable mean. – Георгий Леонидович Стависский Nov 26 '23 at 16:22

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