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About dual one-dimensional oscillator and Coulomb-like theories

We present a mathematically rigorous quantum-mechanical treatment of a one-dimensional nonrelativistic quantum dual theories (with oscillator and Coulomb-like potentials) 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.