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Modern Physics Letters A

ALGEBRAIC APPROACH TO THE POSITION-DEPENDENT MASS SCHRÖDINGER EQUATION FOR A SINGULAR OSCILLATOR

Shi‐Hai Dong, José Juan Peña, C. Pacheco‐García, Jesús García‐Ravelo

Published May 7, 2007• 75 citations
Publisher

Abstract

We construct a singular oscillator Hamiltonian with a position-dependent effective mass. We find that an su(1, 1) algebra is the hidden symmetry of this quantum system and the isospectral potentials V(x) depend on the different choices of the m(x). The complete solutions are also presented by using this Lie algebra.
DOI
10.1142/s0217732307021470

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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.