Unfortunately, it won't work. You'll run into the same kind of problems described in in the original question. In fact, it has to do the principal parts @ vanhees71.
The wave function has to satisfy
$$\psi'(0_+) - \psi'(0_-) = - 2mV_0\psi(0)/\hbar^2.$$ Using your expression,
$$\psi'(0_+) = N_k \sqrt{1-R_k} ik, \quad \psi'(0_-) = N_k(1-\sqrt{R_k}) ik, \quad \psi(0) = N_k \sqrt{1-R_k}.$$ Substituting this in the above condition leads to
$$1 - \frac{1 -...
@Isaac0427
really appreciate your effort. While the wave function you gave is certainly proper, the one I gave is also legitimate. yours can be obtained by combining an odd and an even parity wave functions mentioned in my question. i would like to have wave functions with definite parity.
Thanks for your suggestion. However, the article deals with bound states and the regularization issue seems present only for D>1. Moreover, such issue does not exist if one looks at a discrete version of the Hamiltonian. Whatever, the wave functions must be orthogonal, as physically required...
Hi, thanks for your responses. I know that they are not normalized to unity, but they can still be normalized to Dirac function, which is what's sought for.