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.

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.

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 Solutions of the time-independent Schrödinger equation by uniformization on the unit circle

2019 ◽  
Vol 18 (1) ◽  
pp. 157-165
Author(s):  
Kazimierz Rajchel
Keyword(s):  
Schrödinger Equation ◽  
Unit Circle ◽  
Riccati Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Logarithmic Derivative ◽  
Quantization Condition ◽  
State Function ◽  
Winding Number ◽  
Quantization Rule

Abstract The idea presented here of a general quantization rule for bound states is mainly based on the Riccati equation which is a result of the transformed, time-independent, one-dimensional Schrödinger equation. The condition imposed on the logarithmic derivative of the ground state function W0 allows to present the Riccati equation as the unit circle equation with winding number equal to one which, by appropriately chosen transformations, can be converted into the unit circle equation with multiple winding number. As a consequence, a completely new quantization condition, which gives exact results for any quantum number, is obtained.

Solution of the Schrödinger equation for bound states in closed form

1982 ◽  
Vol 26 (1) ◽  
pp. 662-664 ◽  
Author(s):  
Edgardo Gerck ◽  
Jason A. C. Gallas ◽  
Augusto B. d'Oliveira
Keyword(s):  
Schrödinger Equation ◽  
Closed Form ◽  
Schrodinger Equation ◽  
Bound States

A class of time periodic Hamiltonians with no bound states

1987 ◽  
Vol 105 (1) ◽  
pp. 37-42
Author(s):  
H. Kaneta
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
All Solutions ◽  
Time Periodic

SynopsisWe generalise the Paley–Wiener closedness theorem and apply it to a class of time periodic Hamiltonians to show that all solutions to the corresponding Schrodinger equation decay.

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