scholarly journals The Approximation Property of a One-Dimensional, Time Independent Schrödinger Equation with a Hyperbolic Potential Well

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1351
Author(s):  
Ginkyu Choi ◽  
Soon-Mo Jung
Keyword(s):  
Schrödinger Equation ◽  
Potential Function ◽  
Schrodinger Equation ◽  
Approximation Property ◽  
Ulam Stability ◽  
Potential Well ◽  
One Dimensional ◽  
Parabolic Potential ◽  
Hyperbolic Potential ◽  
The One

A type of Hyers–Ulam stability of the one-dimensional, time independent Schrödinger equation was recently investigated; the relevant system had a parabolic potential wall. As a continuation, we proved a type of Hyers–Ulam stability of the time independent Schrödinger equation under the action of a specific hyperbolic potential well. One of the advantages of this paper is that it proves a type of Hyers–Ulam stability of the Schrödinger equation under the condition that the potential function has singularities.

scholarly journals Perturbation of One-Dimensional Time Independent Schrödinger Equation With a Symmetric Parabolic Potential Wall

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1089 ◽  
Author(s):  
Soon-Mo Jung ◽  
Byungbae Kim
Keyword(s):  
Schrödinger Equation ◽  
Potential Barrier ◽  
Schrodinger Equation ◽  
Ulam Stability ◽  
One Dimensional ◽  
Parabolic Potential ◽  
Wall Potential ◽  
Finite Height ◽  
The One

The first author has recently investigated a type of Hyers-Ulam stability of the one-dimensional time independent Schrödinger equation when the relevant system has a rectangular potential barrier of finite height. In the present paper, we will investigate a type of Hyers-Ulam stability of the Schrödinger equation with the symmetric parabolic wall potential.

scholarly journals Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier

2020 ◽  
Vol 18 (1) ◽  
pp. 1413-1422
Author(s):  
Soon-Mo Jung ◽  
Ginkyu Choi
Keyword(s):  
Schrödinger Equation ◽  
Potential Barrier ◽  
Schrodinger Equation ◽  
Applied Mathematics ◽  
Finite Depth ◽  
Ulam Stability ◽  
Potential Well ◽  
One Dimensional ◽  
The One ◽  
Continuous Work

Abstract In Applied Mathematics Letters 74 (2017), 147–153, the Hyers-Ulam stability of the one-dimensional time-independent Schrödinger equation was investigated when the relevant system has a potential well of finite depth. As a continuous work, we prove in this paper a type of Hyers-Ulam stability of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier of {V}_{0} in height and 2c in width, where {V}_{0} is assumed to be greater than the energy E of the particle under consideration.

Eigenvalues of the Potential Function V=z4±Bz2 and the Effect of Sixth Power Terms

1970 ◽  
Vol 24 (1) ◽  
pp. 73-80 ◽  
Author(s):  
Jaan Laane
Keyword(s):  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Potential Function ◽  
Schrodinger Equation ◽  
Harmonic Potential ◽  
Reduced Form ◽  
Approximation Formula ◽  
One Dimensional ◽  
Ring Puckering ◽  
The One

The one-dimensional Schrödinger equation in reduced form is solved for the potential function V = z4+ Bz2 where B may be positive or negative. The first 17 eigenvalues are reported for 58 values of B in the range −50⩽ B⩽100. The interval of B between the tabulated values is sufficiently small so that the eigenvalues for any B in this range can be found by interpolation. At the limits of the range of B the potential function approaches that of a harmonic oscillator with only small anharmonicity. The effect of a small Cz6 term on this potential is studied and it is concluded that a previously reported approximation formula is quite applicable but only for positive values of B. The success of the quartic—harmonic potential function for the analysis of the ring-puckering vibration is shown; it is also demonstrated that the same potential serves as a useful approximation for many other systems, especially those of the double minimum type.

A TWELFTH-ORDER FOUR-STEP FORMULA FOR THE NUMERICAL INTEGRATION OF THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION

2003 ◽  
Vol 14 (08) ◽  
pp. 1087-1105 ◽  
Author(s):  
ZHONGCHENG WANG ◽  
YONGMING DAI
Keyword(s):  
Schrödinger Equation ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Numerical Test ◽  
New Method ◽  
Step Method ◽  
One Dimensional ◽  
The Stability ◽  
The One ◽  
Order Four

A new twelfth-order four-step formula containing fourth derivatives for the numerical integration of the one-dimensional Schrödinger equation has been developed. It was found that by adding multi-derivative terms, the stability of a linear multi-step method can be improved and the interval of periodicity of this new method is larger than that of the Numerov's method. The numerical test shows that the new method is superior to the previous lower orders in both accuracy and efficiency and it is specially applied to the problem when an increasing accuracy is requested.

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