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

2016 ◽  
Vol 31 (33) ◽  
pp. 1650177 ◽  
Author(s):  
A. M. Ishkhanyan
Keyword(s):  
Exact Solution ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Hypergeometric Functions ◽  
Fundamental Solutions ◽  
Lambert W Function ◽  
Confluent Hypergeometric Functions ◽  
Heun Function ◽  
The One

We introduce two potentials explicitly given by the Lambert-W function for which the exact solution of the one-dimensional stationary Schrödinger equation is written through the first derivative of a double-confluent Heun function. One of these potentials is a singular potential that behaves as the inverse square root in the vicinity of the origin and vanishes exponentially at the infinity. The exact solution of the Schrödinger equation for this potential is given through fundamental solutions each of which presents an irreducible linear combination of two confluent hypergeometric functions. Since the potential is effectively a short-range one, it supports only a finite number of bound states.

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

2018 ◽  
Vol 73 (5) ◽  
pp. 407-414 ◽  
Author(s):  
Tigran A. Ishkhanyan ◽  
Vladimir P. Krainov ◽  
Artur M. Ishkhanyan
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Hypergeometric Functions ◽  
Energy Levels ◽  
Hermite Functions ◽  
Exact Equation ◽  
Square Root ◽  
Centrifugal Barrier ◽  
Confluent Hypergeometric Functions

AbstractWe 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. The potential involves an attractive inverse square root term ~x−1/2 with arbitrary strength and a repulsive centrifugal barrier core ~x−2 with the strength fixed to a constant. This is a potential well defined on the half-axis. Each of the fundamental solutions composing the general solution of the Schrödinger equation is written as an irreducible linear combination, with non-constant coefficients, of two confluent hypergeometric functions. We present the explicit solution in terms of the non-integer order Hermite functions of scaled and shifted argument and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and approximate that by a highly accurate transcendental equation involving trigonometric functions. Finally, we construct an accurate approximation for the bound-state energy levels.

The second Exton potential for the Schrödinger equation

2019 ◽  
Vol 34 (24) ◽  
pp. 1950195
Author(s):  
Artur M. Ishkhanyan ◽  
Jacek Karwowski
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Fractional Power ◽  
Fundamental Solutions ◽  
Transcendental Equation ◽  
Quantization Condition ◽  
Hermite Functions ◽  
Energy Quantization ◽  
Half Axis

Analytical solutions of the Schrödinger equation with a singular, fractional-power potential, referred to as the second Exton potential, are derived and analyzed. The potential is defined on the positive half-axis and supports an infinite number of bound states. It is conditionally integrable and belongs to a biconfluent Heun family. The fundamental solutions are expressed as irreducible linear combinations of two Hermite functions of a scaled and shifted argument. The energy quantization condition results from the boundary condition imposed at the origin. For the exact eigenvalues, which are solutions of a transcendental equation involving two Hermite functions, highly accurate approximation by simple closed-form expressions is derived. The potential is a good candidate for the description of quark–antiquark interaction.

THE SOLUTION OF THE SECOND PÖSCHL–TELLER LIKE POTENTIAL BY NIKIFOROV–UVAROV METHOD

2010 ◽  
Vol 19 (01) ◽  
pp. 123-129 ◽  
Author(s):  
M. G. MIRANDA ◽  
GUO-HUA SUN ◽  
SHI-HAI DONG
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Hypergeometric Functions ◽  
Jacobi Functions ◽  
Uvarov Method

The bound states of the Schrödinger equation for a second Pöschl–Teller like potential are obtained exactly using the Nikiforov–Uvarov method. It is found that the solutions can be explicitly expressed in terms of the Jacobi functions or hypergeometric functions. The complicated normalized wavefunctions are found.

A Sturm-Liouville Equation: The Time-Independent One-Dimensional Schrödinger Equation

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Schrödinger Equation ◽  
Liouville Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Classical Case ◽  
Liouville Theory ◽  
Weyl’S Theorem ◽  
One Dimensional ◽  
Sturm Liouville ◽  
The One

This chapter examines the mathematical properties of the time-independent one-dimensional Schrödinger equation as they relate to Sturm-Liouville problems. The regular Sturm-Liouville theory was generalized in 1908 by the German mathematician Hermann Weyl on a finite closed interval to second-order differential operators with singularities at the endpoints of the interval. Unlike the classical case, the spectrum may contain both a countable set of eigenvalues and a continuous part. The chapter first considers the one-dimensional Schrödinger equation in the standard dimensionless form (with independent variable x) and various relevant theorems, along with the proofs, before discussing bound states, taking into account bound-state theorems and complex eigenvalues. It also describes Weyl's theorem, given the Sturm-Liouville equation, and looks at two cases: the limit point and limit circle. Four examples are presented: an “eigensimple” equation, Bessel's equation of order ? greater than or equal to 0, Hermite's equation, and Legendre's equation.

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