Abstract
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 {V}_{0} in height and 2 c 2c in width, where V 0 {V}_{0} is assumed to be greater than the energy E of the particle under consideration.
Highlights
For an open subinterval I = (a, b) of with −∞ ≤ a < b ≤ +∞ and for an integer n > 0, we are going to define the Hyers-Ulam stability of the linear differential equation of nth order (y(n), y(n−1),..., y′, y, x) = 0, (1)where y : I → is an n times continuously differentiable function
We say that the differential equation (1) satisfies the Hyers-Ulam stability if the following statement is true for each ε > 0: for any n times continuously differentiable function y : I → satisfying the differential inequality
We investigate a type of Hyers-Ulam stability of the one-dimensional time-independent Schrödinger equation (3) for the barrier potential with height V0 and width 2c, where 0 < E < V0
Summary
In a later paper [9], the HyersUlam stability of the one-dimensional time-independent Schrödinger equation. It is possible to apply the one-dimensional Schrödinger equation to analyze the state associated with particles reflected by rectangular potential barriers, but this has some distance from the subject of this paper. We consider the one-dimensional time-independent Schrödinger equation (3), where ψ : → is the wave function, V is a rectangular potential barrier, ħ is the reduced Planck constant, m is the mass of the particle, and E is the energy of the particle. We investigate a type of Hyers-Ulam stability of the one-dimensional time-independent Schrödinger equation (3) for the barrier potential with height V0 and width 2c, where 0 < E < V0. Since the main results of this paper do not fully meet the condition for the Hyers-Ulam stability, instead of saying that the one-dimensional time-independent Schrödinger equation (3) satisfies the Hyers-Ulam stability, we say that the Schrödinger equation with rectangular potential barrier satisfies a type of HyersUlam stability
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