Bogolyubov Institute for Theoretical Physics,

14-B Metrologichna Str., 03143 Kyiv, UKRAINE

E-mail: azolo@gluk.org, az@imm.dtu.dk

**Countable sets of self-adjoint extensions of the Schroedinger
operator with point dipole interaction**

**Abstract:**

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.