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The full spectrum and eigenfunctions of the quantum version of a nonlinearoscillator defined on an N-dimensional space with nonconstant curvature arerigorously found. Since the underlying curved space generates aposition-dependent kinetic energy, three different quantization prescriptionsare worked out by imposing that the maximal superintegrability of the systemhas to be preserved after quantization. The relationships among these threeSchroedinger problems are described in detail through appropriate similaritytransformations. These three approaches are used to illustrate differentfeatures of the quantization problem on N-dimensional curved spaces or,alternatively, of position-dependent mass quantum Hamiltonians. This quantumoscillator is, to the best of our knowledge, the first example of a maximallysuperintegrable quantum system on an N-dimensional space with nonconstantcurvature.

Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability