In this paper, we present an analysis of the equation ẍ−(1/2x)ẋ2+2ω2x−1/8x=0, where ω > 0 and x = x(t) is a real-valued variable. We first discuss the appearance of this equation from a position-dependent-mass scenario in which the mass profile goes inversely with x, admitting a singularity at x = 0. The associated potential is also singular at x = 0, splitting the real axis into two halves, i.e., x > 0 and x < 0. The dynamics is exactly solvable for both the branches and so for definiteness, we stick to the x > 0 branch. Performing a canonical quantization in the position representation and upon employing the ordering strategy of the kinetic-energy operator due to von Roos, we show that the problem is isospectral to the isotonic oscillator. Thus, the quantum spectrum consists of an infinite number of equispaced levels. The spacing between the energy levels is found to be insensitive to the specific choices of the ambiguity parameters that are employed for ordering the kinetic-energy operator à la von Roos.

Quantum Mechanics in Curved Configurational Space Different approaches are compared to formulation of quantum mechanics of a particle on the curved spaces. At first, the canonical, quasi-classical and path integration formalisms are considered for quantization of geodesic motion on the Rimannian configurational spaces. A unique rule of ordering of operators in the canonical formalism and a unique definition of the path integral are established and, thus, a part of ambiguities in the quantum counterpart of geodesic motion is removed. A geometric interpretation is proposed for non-invariance of the quantum mechanics on coordinate transformations. An approach alternative to the quantization of geodesic motion is surveyed, which starts with the quantum theory of a neutral scalar field. Consequences of this alternative approach and the three formalisms of quantization are compared. In particular, the field theoretical approach generates a deformation of the canonical commutation relations between coordinates and momenta of a prticle. A possible cosmological consequence of the deformation is presented in short.

Ordering ambiguity versus representation In this work we show that the ordering ambiguity on quantization depends on the representation choice. This property is then used to solve unambiguously some particular systems. Finally, we speculate on the consequences for more involved cases. Published as: Journal of Physics A: Math. Gen., vol. 39, pages 203-208 (2006) DOI: 10.1088/0305-4470/39/1/014 arXiv categories: quant-ph

This is possible because the introduction of a minimum length comes with an ordering ambiguity much like the ordering ambiguity that arises with the introduction of hbar in the process of quantization. Published as: Phys.Rev. D71 (2005) 023503 DOI: 10.1103/PhysRevD.71.023503 arXiv categories: astro-ph gr-qc hep-th