About dual two-dimensional oscillator and Coulomb-like theories on a   plane
  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. 
G. V. Grigoryan, R. P. Grigoryan, I. V. Tyutin
math-ph math.MP quant-ph