  The quantum version of a non-linear oscillator, previouly analyzed at the
classical level, is studied. This is a problem of quantization of a system with
position-dependent mass of the form $m={(1+\lambda x^2)}^{-1}$ and with a
$\la$-dependent nonpolynomial rational potential. This $\la$-dependent system
can be considered as a deformation of the harmonic oscillator in the sense that
for $\la\to 0$ all the characteristics of the linear oscillator are recovered.
Firstly, the $\la$-dependent Schr\"odinger equation is exactly solved as a
Sturm-Liouville problem and the $\la$-dependent eigenenergies and
eigenfunctions are obtained for both $\la>0$ and $\la<0$. The $\la$-dependent
wave functions appear as related with a family of orthogonal polynomials that
can be considered as $\la$-deformations of the standard Hermite polynomials. In
the second part, the $\la$-dependent Schr\"odinger equation is solved by using
the Schr\"odinger factorization method, the theory of intertwined Hamiltonians
and the property of shape invariance as an approach. Finally, the new family of
orthogonal polynomials is studied. We prove the existence of a $\la$-dependent
Rodrigues formula, a generating function and $\la$-dependent recursion
relations between polynomials of different orders.
