EUROPEAN PHYSICAL JOURNAL PLUS, vol.137, no.5, 2022 (SCI-Expanded)
We extended exactly solvable model of a nonrelativistic quantum linear harmonic oscillator with a position-dependent mass M (x) = a(2)m(0)/a(2)+x(2) to the case where an external homogeneous gravitational field is applied. To describe this system, we use a generalized free quantum Hamiltonian with position-dependent mass, which includes all possible orderings of the momentum (p) over cap = -i (h) over bar partial derivative(x) and mass function M(x) operators that do not commute with each other. As a result, the frequency of the oscillator is renormalized. The square of the renormalized frequency can take on the values Omega(2) > 0, Omega(2) = 0 and Omega(2) < 0. We show that the problem is still exactly solvable and the analytic expression of the wave functions of the stationary states is expressed by means of pseudo-Jacobi polynomials, too. Despite the presence of an external homogeneous field, the number of energy levels remains finite. We have also shown that under the limit a -> infinity the system in the case of Omega(2) > 0 recovers the known nonrelativistic quantum linear harmonic oscillator with constant mass in an external gravitational field and discussed some properties of the generalized quantum Hamiltonian.