  We present a mathematically rigorous quantum-mechanical treatment of a
two-dimensional nonrelativistic quantum dual theories (with oscillator and
Coulomb like potentials) on a plane and compare their spectra and the sets of
eigenfunctions. All self-adjoint Schrodinger operators for these theories are
constructed and rigorous solutions of the corresponding spectral problems are
presented. The first part of the problem is solved by using a method of
specifying s.a. extensions by (asymptotic) s.a. boundary conditions. Solving
spectral problems, we follow the Krein's method of guiding functionals. We
show, that there is one to one correspondence between the spectral points of
dual theories in the planes energy-coupling constants not only for discrete,
but also for continuous spectra.
