The δ(x) Dirac delta is not the only "point-supported" potential that we can integrate; in principle all their derivatives δ′,δ″ exist also, do they?
If yes, can we look for bound states in any of these \delta'^{(n)}(x) potentials? Are there explicit formulae for them (and for the scattering states)?
To be more precise, I am asking for explicit solutions of the 1D Schroedinger equation with point potential,
- {\hbar^2 \over 2m} \Psi_n''(x) + a \ \delta'^{(n)}(x) \Psi(x) \ = E_n \Psi_n(x)
I should add that I have read at least of three set of boundary conditions that are said to be particular solutions:
- \Psi'(0^+)-\Psi'(0^-)= A \Psi(0) with \Psi(0) continuous, is the zero-th derivative case, the "delta potential".
- \Psi(0^+)-\Psi(0^-)= B \Psi'(0) with \Psi'(0) continuous, was called "the delta prime potential" by Holden.
- \lambda \Psi'(0^+)=\Psi'(0^-) and \Psi(0^+)=\lambda\Psi(0^-) simultaneusly, was called "the delta prime potential" by Kurasov
The zero-th derivative case, V(x)=a \delta(x) is a typical textbook example, pretty nice because it only has a bound state, for negative a, and it acts as a kind of barrier for positive a. So it is interesting to ask for other values of n and of course for the general case and if it offers more bound states or other properties. Is it even possible to consider n beyond the first derivative?
Related questions
(If you recall a related question, feel free to suggest it in the comments, or even to edit directly if you have privileges for it)
For the delta prime, including velocity-dependent potentials, the question has been asked in How to interpret the derivative of the Dirac delta potential?
In the halfline r>0, the delta is called "Fermi Pseudopotential". As of today I can not see questions about it, but Classical limit of a quantum system seems to be the same potential.
A general way of taking with boundaring conditions is via the theory of self-adjoint extensions of hermitian operators. This case is not very different of the "particle in 1D box", question Why is \psi = A \cos(kx) not an acceptable wave function for a particle in a box? A general take was the question Physical interpretation of different selfadjoint extensions A related but very exotic question is What is the relation between renormalization and self-adjoint extension? because obviosly the point-supported interacctions have a peculiar scaling
Comments
Of course upgrading distributions to look as operators in L^2 is delicate, and it goes worse for derivatives of distributions when you consider its evaluation <\phi | \rho(x) \psi>. Consider the case \rho(x) = \delta'(x) = \delta(x) {d\over dx}. Should the derivative apply to \psi only, or to the product \phi^*\psi?
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