The Dirac comb with one $\delta$-function removed

Question

Has anyone ever encountered the Dirac comb/Shah function with one removed $\delta$-function, $$ V(x)=\frac{\hbar^2\kappa}{m}\sum_{n\neq0}\delta(x-an), $$ in any literature? I want to find the solution of the Schrödinger equation with such potential but currently, I'm experiencing some difficulties. I believe that this potential describes the lattice defect so I tried to find any appearance of it in the corresponding literature so I could get some ideas, but I failed.

Answer 1

This is an interesting, and likely solvable, generalization of scattering from a delta -potential. I would suggest formulating it differently:

• Taking unmodified Dirac comb as the potential for calculating the Basis functions (i.e., the Bloch waves): $$ V(x) = \lambda\sum_{n=-\infty}^{+\infty}\delta(x-an) $$
• and adding an impurity potential for $n=0$: $$ V_{imp} = \lambda_1\delta(x) $$ This way the problem is more general, since the impurity can have arbitrary potential strength, whereas the case suggested in the question corresponds $\lambda_1=-\lambda$, which is a vacancy rather than impurity. Moreover, in the limit $\lambda \rightarrow 0$ one should recover the solution for plane waves scattered by a delta-potential.

A very similar, although superficially looking very different, is a problem of an impurity in the tight-binding Hamiltonian, which actually corresponds to the Dirac comb with negative $\lambda$ and negative energies: $$ H_0=\sum_{i}\left(tc^\dagger_i c_{i+1} + h.c.\right),\\ H_{imp} = \epsilon_0 c^\dagger_ic_i. $$ This one is solved in many books.