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

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.


Introduction
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 where → y I : is an n times continuously differentiable function. We say that the differential equation (1) satisfies (or has) 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 Obloza demonstrated the Hyers-Ulam stability of the linear differential equation ′( ) + ( ) ( ) = ( ) y x g x y x r x . Thereafter, a number of mathematicians have dealt with this subject (see [6][7][8][9][10][11][12][13]).
The Hyers-Ulam stability of the linear inhomogeneous differential equation of second order ″( ) + ′( ) + ( ) = ( ) y x αy x βy x r x (2) was studied in [8], where α and β are complex numbers (see also [14,15]). In a later paper [9], 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. The ideas of papers [8,9] have a great influence on the present paper. The Schrödinger equation based on postulates of quantum mechanics is a time-dependent equation which yields a time-independent equation that is useful for calculating energy eigenvalues. It is also 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.
In 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. Indeed, we investigate a type of Hyers-Ulam stability of the one-dimensional time-independent Schrödinger equation (3) for the barrier potential with height V 0 and width c 2 , where < < E V 0 0 . 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 Hyers-Ulam stability.

Preliminaries
We denote by λ and μ the roots of characteristic equation, + + = x αx β 0 2 , which is associated with Eq. (2). We introduce in the following lemma a special version of [16,Theorem 3.2] It is to be noted that x 0 can be one of the endpoints of the interval I in the previous lemma because ≤ | | ≤ β t 0 sin 1 holds for any real number t.
Proof. For any ∈ x , we define is an even function with respect to t, we have

A type of Hyers-Ulam stability of the Schrödinger equation
In this section, we investigate a type of Hyers-Ulam stability of the one-dimensional time-independent Schrödinger equation (3), where the potential function → V : is given by where < < E V 0 0 (see the following picture).
In quantum mechanics, the problem of rectangular potential barrier is a typical means of explaining the phenomenon of quantum tunneling and quantum reflection.
Throughout this paper, we investigate a type of Hyers-Ulam stability of the one-dimensional timeindependent Schrödinger equation (3) with the potential function → V : defined by (4). The characteristic roots λ and μ of the one-dimensional time-independent Schrödinger equation (3) are given by In the following theorem, we assume that the function → ψ : is the piecewise twice continuously differentiable function, which means that each of As we see in Eq.
, is strongly affected by the value of x. Hence, we call the phenomenon observed in the following theorem a type of Hyers-Ulam stability.
, , then there exists a piecewise twice continuously differentiable function → ϕ : , which is a solution to the one-dimensional time-independent Schrödinger equation Proof. It is to be proved that there exists a piecewise twice continuously differentiable solution ϕ : to the Schrödinger equation (3) that satisfies inequality (6). First, in view of Lemmas 2.1 and 2.2, there exists a twice continuously differentiable function and Similarly, there exists a twice continuously differentiable function and for all > x c.
The solution ϕ 1 to (7) and the solution ϕ 3 to (9) are given by for all ∈ (− ) x c c , . Indeed, ϕ 2 has the form where A 2 and B 2 are some complex numbers, where we let = an arbitrary real number for , for , an arbitrary real number for , for . Then ϕ is a piecewise twice continuously differentiable solution to the one-dimensional time-independent Schrödinger equation (3). Moreover, it follows from (8), (10), and (12) that inequality (6) holds for all is called a piecewise twice continuously differentiable function provided that each of | (−∞ − ) ψ c , , | (− ) ψ c c , , and | ( +∞) ψ c, is twice continuously differentiable. In the following theorem, under the conditions of continuity for the related function and its derivative, we prove a type of Hyers-Ulam stability of the one-dimensional time-independent Schrödinger equation (3) for the class of piecewise twice continuously differentiable functions. We note that in most problems in quantum mechanics, the continuity conditions of the wave function and its derivative are necessary.
Theorem 3.2. Let E and V 0 be real numbers with < < E V 0 0 . Assume that ≠ + cω n π π 0 2 for any ∈ n . Given > ε 0, if a piecewise twice continuously differentiable function → ψ : , and if ψ is continuous, then there exists a piecewise twice continuously differentiable function → ϕ : , which is a solution to the one-dimensional time-independent Schrödinger equation , , such that ϕ satisfies each of the following boundary conditions (i) ϕ is continuous on ; (ii) ′ ϕ is continuous on and such that where B 1 and B 3 are given in (11), A 2 and B 2 are formulated in (13), and * A 2 and * B 2 are given in (21).
Proof. It is obvious that the proof of Theorem 3.1 is also valid for this theorem. We therefore use the proof of Theorem 3.1 as it is. Moreover, we refer to (13) and consider the following function (− ) → * ϕ c c : , where * A 2 and * B 2 are some complex numbers. We obviously know that * ϕ 2 is a particular solution to the Schrödinger equation (3) for − < < c x c. Let us define the function → ϕ : by

(15)
Then ϕ is a piecewise twice continuously differentiable solution to the one-dimensional time-independent Schrödinger equation (3). By the boundary condition (i), if we impose continuities on ϕ 1 , * ϕ 2 , and ϕ 3 at = − x c and = x c, then it follows from (11) that Moreover, by (11) and the boundary condition (ii), if we impose continuities on ′ ϕ 1 , ′ * ϕ 2 , and ′ ϕ 3 at = − x c and = x c, then we get In the aforementioned four equations, if we perform a long calculation to eliminate * A 2 and * B 2 from them, we get the following: where we set  If we substitute the expressions in (20) for A 1 and A 3 in (16) and (17), then we get  Furthermore, by using the last two equations, we obtain So far, we have proved that the one-dimensional time-independent Schrödinger equation (3) has the piecewise twice continuously differentiable solution ϕ that satisfies the boundary conditions (i) and (ii).
Next, by the continuity of ψ at = − x c, it follows from (8), (11), and (15) that  where B 1 and B 3 are given in (11), A 2 and B 2 are formulated in (13), and * A 2 and * B 2 are given in (21). □ By adding the evenness condition to the set of boundary conditions in Theorem 3.2, we achieve the following corollary.
for all > x c, i.e., ϕ 3 satisfies the one-dimensional time-independent Schrödinger equation (9) and inequality (10) for any > x c because of the evenness of ψ and by (8). By (11), (14), and (20), we get for all > x c. Hence, by (21) and the last equality, and by imposing the evenness on ψ and * ϕ 2 , we obtain that  As we did in the proof of Theorem 3.2, we define the function → ϕ : by using formula (15). Then ϕ is a piecewise twice continuously differentiable solution to the one-dimensional time-independent Schrödinger equation (3) and ϕ satisfies inequality (23) and conditions (i), (ii), and (iii). □

Discussion
We assume that there is an isolated system, which can be described by the one-dimensional time-independent Schrödinger equation (3). Knowing the general solution and initial conditions of the Schrödinger equation determines the past, present, and future of this system completely. So we can say that this system is a predictable one. However, sometimes due to some external disturbances, the system is not determined by Schrödinger equation (3), but it can only be explained by using inequality such as (5). In this case, we cannot accurately predict the future of the disturbed system.
Although the system cannot be accurately predicted due to external disturbances, we say that if the future of the disturbed system follows the exact solution of the Schrödinger equation within a certain error bound, the system (or the Shrödinger equation) has the Hyers-Ulam stability. But if the error bound depends on the value of x, such as inequality (6), we say that the system (or the Schrödinger equation) does not have the Hyers-Ulam stability. The main result obtained in this paper is an example of this phenomenon and the resonance is a typical example.
The above argument is a quotation from the paper [18], and it illustrates how important the Hyers-Ulam concept is.
In this paper, we have not demonstrated the Hyers-Ulam stability of the Schrödinger equation. Instead, we could prove a type of Hyers-Ulam stability, a weaker result than the Hyers-Ulam stability. We have hereby demonstrated that there is an exact solution to the Schrödinger equation near every approximate solution to the equation.

Conclusion
In Section 3, we proved a type of Hyers-Ulam stability of the one-dimensional time-independent Schrödinger equation (3). Because the difference between the approximate solution ψ and the exact solution ϕ of the one-dimensional time-independent Schrödinger equation (3) is strongly influenced by x, we have not proved in Theorem 3.1, Theorem 3.2, or in Corollary 3.3 the Hyers-Ulam stability of the one-dimensional time-independent Schrödinger equation in a strict sense when the potential is a rectangular barrier and < < E V 0 0 . Therefore, instead of saying that we have proved the Hyers-Ulam stability in this paper, we say that we have proved a type of Hyers-Ulam stability.
It is necessary to improve Lemma 2.1 first to demonstrate the exact Hyers-Ulam stability of the Schrödinger equation (3) with a rectangular potential barrier. But it seems that this work will be very difficult. Therefore, instead of attempting this improvement in this paper, we intend to leave this improvement as an open problem.
We now introduce the detailed process of constructing the exact solution ϕ to the Schrödinger equation, which repeatedly appears in the main theorems of this paper.
Let E and V 0 be real numbers with < < E V 0 0 . Assume that the potential function → V : is given by formula (4) and that ≠ + cω n π