Discretization of Natanzon potentials

Abstract

We show that the Natanzon family of potentials is necessarily dropped into a restricted set of distinct potentials involving a fewer number of independent parameters if the potential term in the Schrödinger equation is proportional to an energy-independent parameter and if the potential shape is independent of both energy and that parameter. In the hypergeometric case only six such potentials exist, all five-parametric. Among these, only two (Eckart, Pöschl-Teller) are independent in the sense that each cannot be derived from the other by specifications of the involved parameters. Discussing the solvability of the Schrödinger equation in terms of the single-confluent Heun functions, we show that in this case there exist in total fifteen seven-parametric potentials, of which independent are nine. Six of the inde-pendent potentials present different generalizations of the hypergeometric or confluent hypergeometric ones, while three others do not possess hypergeometric sub-potentials. The result for the double- and bi-confluent Heun equations produces the three independent double- and five independent bi-confluent six-parametric Lamieux-Bose potentials, and the general five-parametric quartic oscillator potential for the tri-confluent Heun equation.

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Cited by in corpus (17)

• Solutions of the bi-confluent Heun equation in terms of the Hermite functions
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• Exact solution of the Schrödinger equation for a short-range exponential potential with inverse square root singularity
• The Lambert-W step-potential - an exactly solvable confluent hypergeometric potential
• A conditionally exactly solvable generalization of the inverse square root potential
• Exact solutions of the sextic oscillator from the bi-confluent Heun equation
• Hermite function solutions of the Schrödinger equation for the sextic oscillator
• Guided modes and terahertz transitions for two-dimensional Dirac fermions in a smooth double-well potential
• Exact solution of the 1D Dirac equation for the inverse-square-root potential $1/\sqrt{x}$
• A conditionally integrable bi-confluent Heun potential involving inverse square root and centrifugal barrier terms
• Semiclassical rates for tunnel ionization from power-law potentials induced by a constant or low-frequency electric field
• A conditionally integrable Schrödinger potential of a bi-confluent Heun class
• A new exactly integrable hypergeometric potential for the Schrödinger equation
• Comment on the paper "Third-harmonic generation investigated by a short-range bottomless exponential potential well" by M. Hu, K. Guo, Q. Yu, Z. Zhang [Superlattices and Microstructures, 122 (2018) 538-547]
• Bi-confluent Heun potentials for a stationary relativistic wave equation for a spinless particle
• A Schrödinger potential involving $x^{\frac{2}{3}}$ and centrifugal-barrier terms conditionally integrable in terms of the confluent hypergeometric functions
• The third five-parametric hypergeometric quantum-mechanical potential