  We present a mathematically rigorous quantum-mechanical treatment of a
two-dimensional nonrelativistic quantum dual theories (with oscillator and
Coulomb like potentials) on pseudosphere and compare their spectra and the sets
of eigenfunctions. We construct all self-adjoint Schrodinger operators for
these theories and represent rigorous solutions of the corresponding spectral
problems. Solving the first part of the problem, we use 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.
