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This is a follow-up for the question Is the helium atom with only a contact interaction between the electrons solvable?. I doubt that there would be a positive answer to that. But I'm interested in any model, which has the following features:

  • Is exactly solvable analytically
  • Can be formulated as a Schrödinger equation in position representation
  • Has bound states in continuum, like helium atom without electron-electron repulsion
  • These bound states can be converted to resonances by addition of an interaction term to the Hamiltonian, without rendering the model unsolvable.

Are there any such models?

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Ruslan
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  • Isn't this what the Fano paper does? If not, why is it unsatisfactory? Or am I misremembering, and it only offers a perturbative theory? – Emilio Pisanty Feb 26 '17 at 14:06
  • @EmilioPisanty I should have been more specific in my question. I'm looking for a model describable as Schrödinger equation in position representation — or at least a method of exact solution, applicable to such a description. Fano's paper, OTOH, works with a limited subset of states, disregarding the others (IIUIC, see equation (1c) and paragraph under it). This doesn't look like exact solution. – Ruslan Feb 26 '17 at 17:00
  • I'm not sure I understand the question, but would the following suffice? Consider a solvable Hamiltonian $H=H(p,q)$ (such as the hydrogen atom). If you introduce a new set of phase-space variables $(q',p')$ and consider $H'=H(p,q)+\frac12p'^2$, then the new eigenenergies are continuously paremetrised by $p'$. Finally, if you add the term $H''=H'+\frac12 q'^2$, the energies become discrete. Does this make any sense to you? – AccidentalFourierTransform Feb 26 '17 at 17:07
  • I don't have access to the paper at the moment, but restricting attention to a relevant substance doesn't degrade the argument much, in my view. It's still a "model", as you require ;-), i.e. it has a Hilbert space, and a hamiltonian on it, so what else do you need? If you then give only approximate solutions, of course, it's a different story. – Emilio Pisanty Feb 26 '17 at 17:09
  • That said, asking for position-representation examples is an interesting enough question, too. My first instinct is to try a shallow square well in 1D but I need to think about it some. – Emilio Pisanty Feb 26 '17 at 17:11
  • @Accident That doesn't really feel like it will model the autoionization process the OP is after. See Ruslan's original question (linked in the linked one) for more context. That said, this one could benefit from making the context clearer. – Emilio Pisanty Feb 26 '17 at 17:13
  • @EmilioPisanty, regarding "what else do you need": I just want to examine an exactly solvable model closely, without distracting approximations. And working with "some" Hilbert space with a Hamiltonian defined as "some" matrix looks a bit too abstract to me for this purpose when I can't map this to a differential equation in an exact way (thus becomes an additional obstacle to good understanding). – Ruslan Feb 26 '17 at 17:31
  • @Ruslan Frankly, I would argue that the distractions and fluff go the other way. (That is, it really is all encoded in the hamiltonian and its matrix elements, and it's insisting on a specific model that becomes an obstacle to good understanding. Plus, the abstract way of looking at it is rather more standard past a certain point, so it's best to get used to it early ;-).) Ultimately it's a matter of taste, though, so to each their own. – Emilio Pisanty Feb 26 '17 at 17:41

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