Countable sets of self-adjoint extensions of the Schroedinger operator with point dipole interaction
The scattering properties of regularizing finite-range potentials constructed in the form of squeezed rectangles, which approximate the derivative of Dirac's delta function are studied in the zero-range limit. Particularly, for a countable set of interaction strength values, a non-zero transmission through the point dipole potential, defined as a limit (in the sense of distributions) of dipole-like sequences of rectangles, is shown to exist as the rectangles are squeezed to zero width. This result is against the actual belief that the point dipole potential in one-dimensional non-relativistic quantum mechanics acts as a totally reflecting wall. For each rectangular-like sequence of finite-range potentials, there exists a one-to-one correspondence between a countable family of interaction strength values and a set of self-adjoint extensions of the non-relativistic kinetic energy operator defined with appropriate boundary conditions at the singularity point.