Dynamical symmetry of a semiconfined harmonic oscillator model with a position-dependent effective mass

Dynamical symmetry algebra for a semiconfined harmonic oscillator model with a position-dependent effective mass is constructed. Selecting the starting point as a well-known factorization method of the Hamiltonian under consideration, we have found three basis elements of this algebra. The algebra defined through those basis elements is a $\mathfrak{su}\left(1,1 \right)$ Heisenberg-Lie algebra. Different special cases and the limit relations from the basis elements to the Heisenberg-Weyl algebra of the non-relativistic quantum harmonic oscillator are discussed, too.