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We present an exact solution of a confined model of the non-relativisticquantum harmonic oscillator, where the effective mass and the angular frequencyare dependent on the position. The free Hamiltonian of the proposed model hasthe form of the BenDaniel--Duke kinetic energy operator. Theposition-dependency of the mass and the angular frequency is such that thehomogeneous nature of the harmonic oscillator force constant $k$ and hence theregular harmonic oscillator potential is preserved. As a consequence thereof, aquantization of the confinement parameter is observed. It is shown that thediscrete energy spectrum of the confined harmonic oscillator withposition-dependent mass and angular frequency is finite, has a non-equidistantform and depends on the confinement parameter. The wave functions of thestationary states of the confined oscillator with position-dependent mass andangular frequency are expressed in terms of the associated Legendre orGegenbauer polynomials. In the limit where the confinement parameter tends to$\infty$, both the energy spectrum and the wave functions converge to thewell-known equidistant energy spectrum and the wave functions of the stationarynon-relativistic harmonic oscillator expressed in terms of Hermite polynomials.The position-dependent effective mass and angular frequency also becomeconstant under this limit.

journal of physics. a, mathematical and theoretical

Exact solution of the position-dependent effective mass and angular  frequency Schr\"odinger equation: harmonic oscillator model with quantized  confinement parameter