Two exactly-solvable confined models of the completely positive
oscillator-shaped quantum well are proposed. Exact solutions of the
position-dependent mass Schr\"odinger equation corresponding to the proposed
quantum well potentials are presented. It is shown that the discrete energy
spectrum expressions of both models depend on certain positive confinement
parameters. The spectrum exhibits positive equidistant behavior for the model
confined only with one infinitely high wall and non-equidistant behavior for
the model confined with the infinitely high wall from both sides. Wavefunctions
of the stationary states of the models under construction are expressed through
the Laguerre and Jacobi polynomials. In general, the Jacobi polynomials
appearing in wavefunctions depend on parameters a and b, but the Laguerre
polynomials depend only on the parameter a. Some limits and special cases of
the constructed models are discussed.Comment: 20 pages, 4 figure