Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 10 (2015), 1 -- 21

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A. Ghose Choudhury and Partha Guha

Abstract. In this survey the role of Jacobi's last multiplier in mechanical systems with a position dependent mass is unveiled. In particular, we map the Liénard II equation x" + f(x)x'2 + g(x) = 0 to a position dependent mass system. The quantization of the Liénard II equation is then carried out using the point canonical transformation method together with the von Roos ordering technique. Finally we show how their eigenfunctions and eigenspectrum can be obtained in terms of associated Laguerre and exceptional Laguerre functions. By employing the exceptional Jacobi polynomials we construct three exactly solvable potentials giving rise to bound-state solutions of the Schrödinger equation.

2010 Mathematics Subject Classification: 58A15; 58A30.
Keywords: Liénard II equation; position-dependent mass; Jacobi last multiplier; Schrödinger equation; exceptional Laguerre equation; exceptional Jacobi polynomial.

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A. Ghose Choudhury Partha Guha
E-mail: E-mail:
Department of Physics, S. N. Bose National Centre for Basic Sciences,
Surendranath College, JD Block, Sector III, Salt Lake,
Mahatma Gandhi Road, Kolkata - 700098, India.
Calcutta-700009, India.