https://doi.org/10.1140/epjp/s13360-022-02715-6
Regular Article
Exactly solvable model of the linear harmonic oscillator with a position-dependent mass under external homogeneous gravitational field
1
Institute of Physics, Azerbaijan National Academy of Sciences, Javid av. 33, 1143, Baku, Azerbaijan
2
Department of Physics, Karadeniz Technical University, 61080, Trabzon, Turkey
3
Department of Theoretical Physics, Baku State University, Z. Khalilov St. 23, 1148, Baku, Azerbaijan
4
Institute for Physical Problems, Baku State University, Z. Khalilov St. 23, 1148, Baku, Azerbaijan
Received:
28
December
2021
Accepted:
10
April
2022
Published online:
3
May
2022
We extended exactly solvable model of a nonrelativistic quantum linear harmonic oscillator with a position-dependent mass 
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 
and mass function 
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 
, 
and 
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 
the system in the case of 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.
© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2022
