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We present a mathematically rigorous quantum-mechanical treatment of atwo-dimensional nonrelativistic quantum dual theories (with oscillator andCoulomb like potentials) on pseudosphere and compare their spectra and the setsof eigenfunctions. We construct all self-adjoint Schrodinger operators forthese theories and represent rigorous solutions of the corresponding spectralproblems. Solving the first part of the problem, we use a method of specifyings.a. extensions by (asymptotic) s.a. boundary conditions. Solving spectralproblems, we follow the Krein's method of guiding functionals. We show, thatthere is one to one correspondence between the spectral points of dual theoriesin the planes energy-coupling constants not only for discrete, but also forcontinuous spectra.

About dual two-dimensional oscillator and Coulomb-like theories on pseudosphere