# Are there any solvable models with bound states in continuum convertible to resonances?

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?

• 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