Quantum singular oscillator with potential controlled by position-dependent mass

An exactly solvable model of a nonrelativistic quantum singular oscillator with a controlled potential is constructed. A position-dependent mass is introduced instead of a constant mass of the quantum system under consideration in such a way that it is possible to purposefully control the shape of the potential, turning it into a quantum well with different depths. It is shown that analytical expressions of the wave functions of such a quantum system are described through 2F1 type hypergeometric functions and Jacobi polynomials, whereas, the discrete energy spectrum corresponding to these analytic solutions is non-equidistant. In a specific limit, when the above-mentioned control of the potential shape is lost, the wave functions accurately restore the wave functions of an ordinary quantum singular oscillator, which are described through generalized Laguerre polynomials. At this limit, the energy spectrum also restores the usual equidistant energy spectrum of a quantum singular oscillator.