# Potentials of the Heun class

@article{Batic2013PotentialsOT, title={Potentials of the Heun class}, author={David Batic and R Williams and Marek Nowakowski}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2013}, volume={46} }

We review different methods of generating potentials such that the one-dimensional Schrödinger equation (ODSE) can be transformed into the hypergeometric equation. We compare our results with previous studies, and complement the subject with new findings. Our main result is to derive new classes of potentials such that the ODSE can be transformed into the Heun equation and its confluent cases. The generalized Heun equation is also considered.

## 25 Citations

Schrödinger potentials solvable in terms of the confluent Heun functions

- Mathematics, Physics
- 2016

We show that if the potential is proportional to an energy-independent continuous parameter, then there exist 15 choices for the coordinate transformation that provide energy-independent potentials…

Semicommuting and Commuting Operators for the Heun Family

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- 2017

We derive the most general families of first- and second-order differential operators semicommuting with the Heun class differential operators. Among these families, we classify all the families that…

A singular Lambert-W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions

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- 2016

We introduce two potentials explicitly given by the Lambert-W function for which the exact solution of the one-dimensional stationary Schrodinger equation is written through the first derivative of a…

Discretization of Natanzon potentials

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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 Lambert-W step-potential – an exactly solvable confluent hypergeometric potential

- Mathematics, Physics
- 2016

A Conditionally Integrable Bi-confluent Heun Potential Involving Inverse Square Root and Centrifugal Barrier Terms

- Mathematics
- 2017

Abstract We present a conditionally integrable potential, belonging to the bi-confluent Heun class, for which the Schrödinger equation is solved in terms of the confluent hypergeometric functions.…

Confluent Heun equation with single added apparent singularity

- Mathematics, Physics2016 Days on Diffraction (DD)
- 2016

Elementary, gauge and Laplace integral symmetries of confluent Heun equation with single added apparent singularity are under consideration. Symmetries from this collection connect solutions of the…

Unified supersymmetric transformations for the harmonic oscillator and its rational extension

- PhysicsEuropean Journal of Physics
- 2020

We discussed transformations of supersymmetric quantum mechanics by considering a general six-parameter function, from which the superpotential and the supersymmetric partner potentials V−(r) and…

Solutions of the bi-confluent Heun equation in terms of the Hermite functions

- Mathematics, Physics
- 2016

## References

SHOWING 1-10 OF 61 REFERENCES

Algebraic treatment of the confluent Natanzon potentials

- Physics
- 2000

Using the so(2,1) Lie algebra and the Baker, Campbell and Hausdorff formulas, the Green's function for the class of the confluent Natanzon potentials is constructed straightforwardly. The bound-state…

Conditionally solvable path integral problems: II. Natanzon potentials

- Mathematics
- 1996

New classes of exactly solvable potentials are discussed within the path integral formalism. They are constructed from the hypergeometric and confluent Natanzon potentials, respectively. It is found…

Heun's equation, generalized hypergeometric function and exceptional Jacobi polynomial

- Mathematics
- 2012

We study Heun's differential equation in the case that one of the singularities is apparent. In particular, we propose a conjecture that solutions of Heun’s equation in this case also satisfy a…

Transformations of Heun's equation and its integral relations

- Mathematics
- 2010

Using the transformation theory for the Heun equation, we find substitutions of variables which preserve the form of the equation for the kernels of integral relations among solutions of the Heun…

A hypergeometric system of the Heun equation and middle convolution

- Mathematics
- 2009

A new method of obtaining integral transformation of the Heun equation is described. The Heun equation is reduced to the so-called hypergeometric system, a linear Fuchsian system of rank 3, and a…

Heun functions and quasi-exactly solvable double-well potentials

- Physics
- 2012

We investigate a type of one-dimensional quasi-exactly solvable double-well potential whose analytical solution can be constructed in terms of the Heun functions. It is shown that for certain special…

Liouville Transformation and Exactly Solvable Schrodinger Equations

- Physics, Mathematics
- 1997

The present paper discusses the connectionbetween exactly solvable Schrodinger equations and theLiouville transformation. This transformation yields alarge class of exactly solvable potentials,…

A class of solvable potentials

- Physics
- 1964

SummaryThe problem of the construction of solvable one-variable Schrödinger potentials is formulated. A class of simple potentials for which the Schrödinger equation can be solved in terms of special…

Examples of Heun and Mathieu functions as solutions of wave equations in curved spaces

- Mathematics
- 2007

We give examples of where the Heun function exists as solutions of wave equations encountered in general relativity. As a new example we find that while the Dirac equation written in the background…

The generalized Heun equation in QFT in curved spacetimes

- Physics
- 2006

In this paper we give a brief outline of the applications of the generalized Heun equation (GHE) in the context of quantum field theory in curved spacetimes. In particular, we relate the separated…