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The linear differential equations with complex constant coefficients and Schrödinger equations
Applied Mathematics Letters 66 (2017) 23–29
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
The linear differential equations with complex constant coefficients and Schrödinger equations Soon-Mo Junga , Jaiok Roh b, * a
Mathematics Section, College of Science and Technology, Hongik University, 30016 Sejong, Republic of Korea b Department of Mathematics, Hallym University, Chuncheon, Kangwon-Do 24252, Republic of Korea
article
info
Article history: Received 19 September 2016 Received in revised form 8 November 2016 Accepted 9 November 2016 Available online 17 November 2016
abstract We investigate some properties of approximate solutions for the second-order inhomogeneous linear differential equations, y ′′ (x) + αy ′ (x) + βy(x) = r(x), with complex constant coefficients. And, as an application of our results, we will see the time independent Schrödinger equations. This paper was motivated by the paper, Li and Shen (2010). © 2016 Elsevier Ltd. All rights reserved.
Keywords: Hyers–Ulam stability Generalized Hyers–Ulam stability Linear differential equation Approximation Schrödinger equation
1. Introduction Throughout this paper, let I = (a, b) be an open interval of R with −∞ ≤ a < b ≤ +∞. For a positive integer n, we will consider the linear differential equation of nth order ( ) F y (n) , y (n−1) , . . . , y ′ , y, x = 0 (1.1) defined on I, where y : I → C is an n times continuously differentiable function. If, for any ε > 0 and n times continuously differentiable function y : I → C satisfying the differential inequality ⏐ ( (n) (n−1) )⏐ ⏐F y , y , . . . , y ′ , y, x ⏐ ≤ ε (1.2) for all x ∈ I, there exists a solution y0 : I → C to the differential equation (1.1) such that |y(x) − y0 (x)| ≤ K(ε) * Corresponding author. E-mail addresses: [email protected] (S. Jung), [email protected] (J. Roh). http://dx.doi.org/10.1016/j.aml.2016.11.003 0893-9659/© 2016 Elsevier Ltd. All rights reserved.
(1.3)
S. Jung, J. Roh / Applied Mathematics Letters 66 (2017) 23–29
24
for all x ∈ I, where K(ε) depends on ε only and satisfies limε→0 K(ε) = 0, then we say that the differential equation (1.1) satisfies (or has) the Hyers–Ulam stability (or the local Hyers–Ulam stability if the domain I is not the whole space R). Obloza seems to be the first author who investigated the Hyers–Ulam stability of linear differential equations (see [1,2]). Obloza studied the linear differential equations of the form, y ′ (x) + g(x)y(x) − r(x) = 0. Thereafter, Alsina and Ger [3] proved that if a differentiable function y : (a, b) → R satisfies the differential inequality |y ′ (x) − y(x)| ≤ ε, then there exists a function y0 : (a, b) → R such that y0′ (x) = y0 (x) and |y(x) − y0 (x)| ≤ 3ε for all x ∈ (a, b). This result of Alsina and Ger was generalized by Takahasi et al. [4]. For the Hyers–Ulam stability of the differential equation y ′ (x) = λy(x), one can also refer [5–12]. Recently, Li, Zada and Faisal [13] obtained very interesting results about the Hyers–Ulam stability of n-th order linear differential equations. In this paper, we consider the linear inhomogeneous differential equation of second order y ′′ (x) + αy ′ (x) + βy(x) = r(x),
(1.4)
where α and β are complex-valued constants. Here, we assume that the function y : I → C is twice continuously differentiable, and r : I → C is a continuous function. The Hyers–Ulam stability of the differential equation (1.4) has been proved under various additional conditions (see [14,15]). Main difference between our result and that in [14,15] is that we obtain the Hyers–Ulam stability of the linear differential equation (1.4) under weaker conditions in comparison with those of [14,15]. So, we can apply our results to the time independent Schr¨ odinger equations. 2. Linear differential equations with constant coefficients In this section, we consider the second-order inhomogeneous linear differential equations (1.4) with complex constant coefficients by generalizing the ideas from [15], where α and β are complex-valued constants and r : I → C is a continuous function. We denote by λ and µ the roots of the characteristic equation x2 + αx + β = 0 corresponding to (1.4) and we set p = ℜ(λ) and q = ℜ(µ). Theorem 2.1. Let I = (a, b) be an open interval with −∞ ≤ a < b ≤ +∞. Assume that both p and q are positive numbers. For a twice continuously differentiable function y : I → C and a continuous function r : I → C, define ( ) ∫ s g(x) = y ′ (x) − µy(x), z(x) = lim g(s)e−λ(s−x) − eλx r(t)e−λt dt (2.1) s→b
for all x ∈ I. In addition, assume that each of ∫ b ∫ b r(t)e−λt dt, z(t)e−µt dt, x
x
x
lim g(s)e−λs ,
s→b
lim y(s)e−µs
s→b
exists for any x ∈ I. Given ε ≥ 0, if y satisfies the inequality ⏐ ′′ ⏐ ⏐y (x) + αy ′ (x) + βy(x) − r(x)⏐ ≤ ε for all x ∈ I, then there exists a solution u : I → C to the differential equation (1.4) such that |y(x) − u(x)| ≤ for all x ∈ I.
ε pq
(2.2)
S. Jung, J. Roh / Applied Mathematics Letters 66 (2017) 23–29
25
Proof . If we differentiate z in (2.1) with respect to x, then we have ) 1( z(x + h) − z(x) h( ) ( ∫ s 1 −λt −λ(s−x−h) λ(x+h) = lim r(t)e dt lim g(s)e −e h→0 h s→b x+h )) ( ∫ s r(t)e−λt dt − lim g(s)e−λ(s−x) − eλx s→b x ( ) ∫ s eλh − 1 −λ(s−x) λx −λt = lim lim g(s)e r(t)e dt −e h→0 s→b h x ∫ x+h 1 r(t)e−λt dt + lim eλ(x+h) h→0 h x = λz(x) + r(x)
z ′ (x) = lim
h→0
(2.3)
for any x ∈ I. Moreover, it follows from (2.1) that g ′ (x) − λg(x) − r(x) = y ′′ (x) + αy ′ (x) + βy(x) − r(x) and hence, by (2.2), we get ⏐ ′ ⏐ ⏐ ⏐ ⏐g (x) − λg(x) − r(x)⏐ = ⏐y ′′ (x) + αy ′ (x) + βy(x) − r(x)⏐ ≤ ε
(2.4)
for each x ∈ I. It follows from (2.1) and (2.4) that for all x ∈ I, we have ⏐ ( )⏐ ∫ s ⏐ ⏐ λx −λs −λx −λt ⏐ − g(x)e − r(t)e dt ⏐⏐ |z(x) − g(x)| = ⏐e lim g(s)e s→b x ⏐∫ s ⏐ ∫ s ⏐ ⏐ ( )′ px −λt = e lim ⏐⏐ g(t)e dt − r(t)e−λt dt⏐⏐ s→b x ∫ xs ⏐ ⏐ −λt ⏐⏐ ′ px ⏐e ⏐⏐g (t) − λg(t) − r(t)⏐dt ≤ e lim s→b x ( ) ε ε ≤ lim 1 − e−p(s−x) ≤ . p s→b p
(2.5)
Now, we define ( ∫ −µ(s−x) µx u(x) = lim y(s)e −e s→b
s
z(t)e
−µt
) dt
(2.6)
x
for any x ∈ I. Then, similar to (2.3), we obtain u′ (x) = µu(x) + z(x),
for all x ∈ I.
So, due to (2.3) and (2.7), we have u′′ (x) + αu′ (x) + βu(x) = r(x),
for all x ∈ I
which implies that u is a solution to the differential equation (1.4).
(2.7)
S. Jung, J. Roh / Applied Mathematics Letters 66 (2017) 23–29
26
Hence, by (2.1), (2.5), and (2.6), we obtain ⏐ ( )⏐ ∫ s ⏐ ⏐ z(t)e−µt dt ⏐⏐ |y(x) − u(x)| = ⏐⏐y(x) − lim y(s)e−µ(s−x) − eµx s→b ⏐ ⏐ ∫x s ⏐ ⏐ ⏐ µx ⏐ −µx −µs −µt ⏐ ⏐ ⏐ ⏐ − y(s)e + z(t)e dt⏐ lim ⏐y(x)e = e s→b x ⏐ ⏐∫ s ∫ s ⏐ ( ) ⏐ −µt ′ ⏐ −µt qx ⏐ dt⏐ y(t)e z(t)e dt − = e lim ⏐ s→b x ⏐ ⏐∫xs ⏐ ( ) ⏐ = eqx lim ⏐⏐ e−µt z(t) − y ′ (t) + µy(t) dt⏐⏐ s→b ⏐ ⏐∫xs ∫ s ⏐ ( ) ⏐ ε −µt qx ⏐ e−qt dt z(t) − g(t) dt⏐⏐ ≤ eqx lim e = e lim ⏐ s→b x s→b p (x ) ε ε −q(σ−x) lim 1 − e , for all x ∈ I. □ ≤ ≤ pq s→b pq
(2.8)
Theorem 2.2. Let I = (a, b) be an open interval with −∞ < a < b < +∞. Assume that p and q are nonnegative numbers. Also, the functions y(x), r(x), g(x) and z(x) satisfy same assumptions with that of Theorem 2.1. Then there exists a solution u : I → C to the differential equation (1.4) such that ⎧ε ⎪ (for p, q > 0), ⎪ ⎪ pq ⎪ ⎨ε (b − a) (for p = 0, q > 0), |y(x) − u(x)| ≤ ⎪q ⎪ ⎪ ⎪ ⎩ ε (b − a)2 (for p = q = 0) 2 for all x ∈ I. Proof . If p, q > 0 then we follow the proof of Theorem 2.1. If p = 0 and q > 0 then for (2.5) we have |z(x) − g(x)| ≤ (b − x)ε for all x ∈ I. Therefore, for (2.8) we obtain ⏐∫ s ⏐ ⏐ ( ) ⏐ |y(x) − u(x)| = eqx lim ⏐⏐ e−µt z(t) − g(t) dt⏐⏐ s→b ∫ xs (b − a)ε ≤ eqx lim e−qt ε(b − t)dt ≤ , s→b x q
for all x ∈ I.
If p = q = 0 then for (2.5) we have |z(x) − g(x)| ≤ (b − x)ε for all x ∈ I. Therefore, for (2.8) we obtain ⏐∫ ⏐ qx |y(x) − u(x)| = e lim ⏐⏐ s→b
≤
s
x
e
−µt
(
⏐ ∫ ) ⏐ z(t) − g(t) dt⏐⏐ ≤ lim s→b
(b − x)2 (b − a)2 ε ≤ ε, 2 2
s
ε(b − t)dt
x
for all x ∈ I.
□
In next Remark, we consider the cases that one of real parts for the roots of the characteristic equation is a negative number. In these cases, we use little different functions for z(x) and u(x).
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S. Jung, J. Roh / Applied Mathematics Letters 66 (2017) 23–29
Remark 2.3. Case 1, p < 0 and q < 0: In this case, we define the functions, g(x) and z(x) as ( ) ∫ x g(x) = y ′ (x) − µy(x), z(x) = lim g(s)eλ(x−s) + eλx r(t)e−λt dt s→a
s
for all x ∈ I. Then, similar to (2.3), we get z ′ (x) = λz(x) + r(x) and in view of (2.2), the inequality (2.4) is also true for all x ∈ I. Moreover, similar to (2.5), we have ⏐ ⏐∫ x ⏐ ( ) ⏐ e−λt r(t) − g ′ (t) + λg(t) dt⏐⏐ |z(x) − g(x)| = epx lim ⏐⏐ s→a ∫ sx ⏐ ⏐ px ≤ e lim e−pt ⏐g ′ (t) − λg(t) − r(t)⏐dt s→a s ) ε( ε ≤− 1 − ep(x−a) ≤ − , for all x ∈ I. p p Next, for the function u(x) in the proof of Theorem 2.1, we define ( ) ∫ x u(x) = lim y(s)eµ(x−s) + eµx z(t)e−µt dt , s→a
for all x ∈ I.
s
Then, by following the process of the proof for Theorem 2.1, one notes that u is a solution to (1.4) and satisfies |y(x) − u(x)| ≤
ε , pq
for all x ∈ I.
Case 2, p < 0 and q > 0: We define the functions, g(x) and z(x) as Case 1 to get |z(x) − g(x)| ≤ −
ε p
for all x ∈ I. And, for the function u(x) in the proof of Theorem 2.1, we define ( ) ∫ s −µ(s−x) µx −µt u(x) = lim y(s)e −e z(t)e dt , for all x ∈ I. s→b
x
Then u is a solution to (1.4) and satisfies |y(x) − u(x)| ≤ −
ε . pq
Case 3, p < 0 and q = 0: In this case, we define the functions, g(x) and z(x) as Case 1 to get |z(x) − g(x)| ≤ −
ε p
for all x ∈ I. And, for the function u(x) in the proof of Theorem 2.1, we define ( ) ∫ x µ(x−s) µx −µt u(x) = lim y(s)e +e z(t)e dt , for all x ∈ I. s→a
s
Then u is a solution to (1.4) and satisfies ε |y(x) − u(x)| ≤ − (b − a), p
for all x ∈ I.
S. Jung, J. Roh / Applied Mathematics Letters 66 (2017) 23–29
28
Remark 2.4. In [15], Li and Shen also obtained the interesting results about the Hyers–Ulam stability of the second-order linear differential equations. Main difference is that they considered the real constant coefficient linear differential equations while we considered the complex constant coefficient linear differential equations. Also, they restricted to have two different positive real roots p and q for the characteristic equations of the linear differential equations. But, in this paper we do not have any restrictions for the roots of the characteristic equations. As a result we can apply our result to the time independent Schr¨ odinger equations. Moreover, they restricted for the bounded domain I = (a, b) while we extended to the unbounded domain −∞ ≤ a < b ≤ ∞ in Theorem 2.1. And with their assumptions we obtained same estimations with that in their paper.
3. Applications We consider the Schr¨ odinger equation for one spatial dimension −
ℏ2 d2 ψ(x) + V ψ(x) = Eψ(x), 2m dx2
(3.1)
where the wave function ψ : (a, b) → C is twice continuously differentiable, V is the potential energy, and E is the total energy of the quantum system. In particular, we assume that V ≤ E. The roots of the characteristic equation of Schr¨ odinger equation (3.1) are √ √ 2m(E − V ) 2m(E − V ) λ = −i and µ = i . ℏ2 ℏ2 Then we know that p = ℜ(λ) = 0 and q = ℜ(µ) = 0. Hence, by Theorem 2.2, if the function ψ satisfies the inequality ⏐ ⏐ ⏐ ℏ2 d2 ψ(x) ⏐ ⏐− ⏐ ≤ ε, for all x ∈ I, (3.2) + V ψ(x) − Eψ(x) ⏐ 2m dx2 ⏐ then there exists a solution ϕ : I → C of the differential equation (3.1) such that |ψ(x) − ϕ(x)| ≤
mε (b − a)2 , ℏ2
for all x ∈ I.
Acknowledgments This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2015R1D1A1A02061826) and (NRF2015R1D1A1A01059467). This work was also supported by 2016 Hongik University Research Fund and Hallym University Research Fund (HRF-201608-004). References
[1] M. Obloza, Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat. 13 (1993) 259–270. [2] M. Obloza, Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.Dydakt. Prace Mat. 14 (1997) 141–146. [3] C. Alsina, R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl. 2 (1998) 373–380. [4] S.-E. Takahasi, T. Miura, S. Miyajima, On the Hyers-Ulam stability of the Banach space-valued differential equation y ′ = λy, Bull. Korean Math. Soc. 39 (2002) 309–315.
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[5] G. Choi, S.-M. Jung, Invariance of Hyers-Ulam stability of linear differential equations and its applications, Adv. Differential Equations 2015 (2015) 14 pages. http://dx.doi.org/10.1186/s13662-015-0617-1. Article no. 277. [6] S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett. 17 (2004) 1135–1140. [7] S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order, II, Appl. Math. Lett. 19 (2006) 854–858. [8] S.-M. Jung, Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients, J. Math. Anal. Appl. 320 (2006) 549–561. [9] T. Miura, H. Oka, S.-E. Takahasi, N. Niwa, Hyers-Ulam stability of the first order linear differential equation for Banach space-valued holomorphic mappings, J. Math. Inequal. 3 (2007) 377–385. [10] D. Popa, I. Ra¸sa, On the Hyers-Ulam stability of the linear differential equation, J. Math. Anal. Appl. 381 (2011) 530–537. [11] D. Popa, I. Ra¸sa, Hyers-Ulam stability of the linear differential operator with non-constant coefficients, Appl. Math. Comput. 219 (2012) 1562–1568. [12] G. Wang, M. Zhou, L. Sun, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett. 21 (2008) 1024–1028. [13] T. Li, A. Zada, S. Faisal, Hyers-Ulam stability of nth order linear differential equations, J. Nonlinear Sci. Appl. 9 (2016) 2070–2075. [14] D.S. Cˆımpean, D. Popa, On the stability of the linear differential equation of higher order with constant coefficients, Appl. Math. Comput. 217 (2010) 4141–4146. [15] Y. Li, Y. Shen, Hyers-Ulam stability of linear differential equations of second order, Appl. Math. Lett. 23 (2010) 306–309.
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
The linear differential equations with complex constant coefficients and Schrödinger equations Soon-Mo Junga , Jaiok Roh b, * a
Mathematics Section, College of Science and Technology, Hongik University, 30016 Sejong, Republic of Korea b Department of Mathematics, Hallym University, Chuncheon, Kangwon-Do 24252, Republic of Korea
article
info
Article history: Received 19 September 2016 Received in revised form 8 November 2016 Accepted 9 November 2016 Available online 17 November 2016
abstract We investigate some properties of approximate solutions for the second-order inhomogeneous linear differential equations, y ′′ (x) + αy ′ (x) + βy(x) = r(x), with complex constant coefficients. And, as an application of our results, we will see the time independent Schrödinger equations. This paper was motivated by the paper, Li and Shen (2010). © 2016 Elsevier Ltd. All rights reserved.
Keywords: Hyers–Ulam stability Generalized Hyers–Ulam stability Linear differential equation Approximation Schrödinger equation
1. Introduction Throughout this paper, let I = (a, b) be an open interval of R with −∞ ≤ a < b ≤ +∞. For a positive integer n, we will consider the linear differential equation of nth order ( ) F y (n) , y (n−1) , . . . , y ′ , y, x = 0 (1.1) defined on I, where y : I → C is an n times continuously differentiable function. If, for any ε > 0 and n times continuously differentiable function y : I → C satisfying the differential inequality ⏐ ( (n) (n−1) )⏐ ⏐F y , y , . . . , y ′ , y, x ⏐ ≤ ε (1.2) for all x ∈ I, there exists a solution y0 : I → C to the differential equation (1.1) such that |y(x) − y0 (x)| ≤ K(ε) * Corresponding author. E-mail addresses: [email protected] (S. Jung), [email protected] (J. Roh). http://dx.doi.org/10.1016/j.aml.2016.11.003 0893-9659/© 2016 Elsevier Ltd. All rights reserved.
(1.3)
S. Jung, J. Roh / Applied Mathematics Letters 66 (2017) 23–29
24
for all x ∈ I, where K(ε) depends on ε only and satisfies limε→0 K(ε) = 0, then we say that the differential equation (1.1) satisfies (or has) the Hyers–Ulam stability (or the local Hyers–Ulam stability if the domain I is not the whole space R). Obloza seems to be the first author who investigated the Hyers–Ulam stability of linear differential equations (see [1,2]). Obloza studied the linear differential equations of the form, y ′ (x) + g(x)y(x) − r(x) = 0. Thereafter, Alsina and Ger [3] proved that if a differentiable function y : (a, b) → R satisfies the differential inequality |y ′ (x) − y(x)| ≤ ε, then there exists a function y0 : (a, b) → R such that y0′ (x) = y0 (x) and |y(x) − y0 (x)| ≤ 3ε for all x ∈ (a, b). This result of Alsina and Ger was generalized by Takahasi et al. [4]. For the Hyers–Ulam stability of the differential equation y ′ (x) = λy(x), one can also refer [5–12]. Recently, Li, Zada and Faisal [13] obtained very interesting results about the Hyers–Ulam stability of n-th order linear differential equations. In this paper, we consider the linear inhomogeneous differential equation of second order y ′′ (x) + αy ′ (x) + βy(x) = r(x),
(1.4)
where α and β are complex-valued constants. Here, we assume that the function y : I → C is twice continuously differentiable, and r : I → C is a continuous function. The Hyers–Ulam stability of the differential equation (1.4) has been proved under various additional conditions (see [14,15]). Main difference between our result and that in [14,15] is that we obtain the Hyers–Ulam stability of the linear differential equation (1.4) under weaker conditions in comparison with those of [14,15]. So, we can apply our results to the time independent Schr¨ odinger equations. 2. Linear differential equations with constant coefficients In this section, we consider the second-order inhomogeneous linear differential equations (1.4) with complex constant coefficients by generalizing the ideas from [15], where α and β are complex-valued constants and r : I → C is a continuous function. We denote by λ and µ the roots of the characteristic equation x2 + αx + β = 0 corresponding to (1.4) and we set p = ℜ(λ) and q = ℜ(µ). Theorem 2.1. Let I = (a, b) be an open interval with −∞ ≤ a < b ≤ +∞. Assume that both p and q are positive numbers. For a twice continuously differentiable function y : I → C and a continuous function r : I → C, define ( ) ∫ s g(x) = y ′ (x) − µy(x), z(x) = lim g(s)e−λ(s−x) − eλx r(t)e−λt dt (2.1) s→b
for all x ∈ I. In addition, assume that each of ∫ b ∫ b r(t)e−λt dt, z(t)e−µt dt, x
x
x
lim g(s)e−λs ,
s→b
lim y(s)e−µs
s→b
exists for any x ∈ I. Given ε ≥ 0, if y satisfies the inequality ⏐ ′′ ⏐ ⏐y (x) + αy ′ (x) + βy(x) − r(x)⏐ ≤ ε for all x ∈ I, then there exists a solution u : I → C to the differential equation (1.4) such that |y(x) − u(x)| ≤ for all x ∈ I.
ε pq
(2.2)
S. Jung, J. Roh / Applied Mathematics Letters 66 (2017) 23–29
25
Proof . If we differentiate z in (2.1) with respect to x, then we have ) 1( z(x + h) − z(x) h( ) ( ∫ s 1 −λt −λ(s−x−h) λ(x+h) = lim r(t)e dt lim g(s)e −e h→0 h s→b x+h )) ( ∫ s r(t)e−λt dt − lim g(s)e−λ(s−x) − eλx s→b x ( ) ∫ s eλh − 1 −λ(s−x) λx −λt = lim lim g(s)e r(t)e dt −e h→0 s→b h x ∫ x+h 1 r(t)e−λt dt + lim eλ(x+h) h→0 h x = λz(x) + r(x)
z ′ (x) = lim
h→0
(2.3)
for any x ∈ I. Moreover, it follows from (2.1) that g ′ (x) − λg(x) − r(x) = y ′′ (x) + αy ′ (x) + βy(x) − r(x) and hence, by (2.2), we get ⏐ ′ ⏐ ⏐ ⏐ ⏐g (x) − λg(x) − r(x)⏐ = ⏐y ′′ (x) + αy ′ (x) + βy(x) − r(x)⏐ ≤ ε
(2.4)
for each x ∈ I. It follows from (2.1) and (2.4) that for all x ∈ I, we have ⏐ ( )⏐ ∫ s ⏐ ⏐ λx −λs −λx −λt ⏐ − g(x)e − r(t)e dt ⏐⏐ |z(x) − g(x)| = ⏐e lim g(s)e s→b x ⏐∫ s ⏐ ∫ s ⏐ ⏐ ( )′ px −λt = e lim ⏐⏐ g(t)e dt − r(t)e−λt dt⏐⏐ s→b x ∫ xs ⏐ ⏐ −λt ⏐⏐ ′ px ⏐e ⏐⏐g (t) − λg(t) − r(t)⏐dt ≤ e lim s→b x ( ) ε ε ≤ lim 1 − e−p(s−x) ≤ . p s→b p
(2.5)
Now, we define ( ∫ −µ(s−x) µx u(x) = lim y(s)e −e s→b
s
z(t)e
−µt
) dt
(2.6)
x
for any x ∈ I. Then, similar to (2.3), we obtain u′ (x) = µu(x) + z(x),
for all x ∈ I.
So, due to (2.3) and (2.7), we have u′′ (x) + αu′ (x) + βu(x) = r(x),
for all x ∈ I
which implies that u is a solution to the differential equation (1.4).
(2.7)
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Hence, by (2.1), (2.5), and (2.6), we obtain ⏐ ( )⏐ ∫ s ⏐ ⏐ z(t)e−µt dt ⏐⏐ |y(x) − u(x)| = ⏐⏐y(x) − lim y(s)e−µ(s−x) − eµx s→b ⏐ ⏐ ∫x s ⏐ ⏐ ⏐ µx ⏐ −µx −µs −µt ⏐ ⏐ ⏐ ⏐ − y(s)e + z(t)e dt⏐ lim ⏐y(x)e = e s→b x ⏐ ⏐∫ s ∫ s ⏐ ( ) ⏐ −µt ′ ⏐ −µt qx ⏐ dt⏐ y(t)e z(t)e dt − = e lim ⏐ s→b x ⏐ ⏐∫xs ⏐ ( ) ⏐ = eqx lim ⏐⏐ e−µt z(t) − y ′ (t) + µy(t) dt⏐⏐ s→b ⏐ ⏐∫xs ∫ s ⏐ ( ) ⏐ ε −µt qx ⏐ e−qt dt z(t) − g(t) dt⏐⏐ ≤ eqx lim e = e lim ⏐ s→b x s→b p (x ) ε ε −q(σ−x) lim 1 − e , for all x ∈ I. □ ≤ ≤ pq s→b pq
(2.8)
Theorem 2.2. Let I = (a, b) be an open interval with −∞ < a < b < +∞. Assume that p and q are nonnegative numbers. Also, the functions y(x), r(x), g(x) and z(x) satisfy same assumptions with that of Theorem 2.1. Then there exists a solution u : I → C to the differential equation (1.4) such that ⎧ε ⎪ (for p, q > 0), ⎪ ⎪ pq ⎪ ⎨ε (b − a) (for p = 0, q > 0), |y(x) − u(x)| ≤ ⎪q ⎪ ⎪ ⎪ ⎩ ε (b − a)2 (for p = q = 0) 2 for all x ∈ I. Proof . If p, q > 0 then we follow the proof of Theorem 2.1. If p = 0 and q > 0 then for (2.5) we have |z(x) − g(x)| ≤ (b − x)ε for all x ∈ I. Therefore, for (2.8) we obtain ⏐∫ s ⏐ ⏐ ( ) ⏐ |y(x) − u(x)| = eqx lim ⏐⏐ e−µt z(t) − g(t) dt⏐⏐ s→b ∫ xs (b − a)ε ≤ eqx lim e−qt ε(b − t)dt ≤ , s→b x q
for all x ∈ I.
If p = q = 0 then for (2.5) we have |z(x) − g(x)| ≤ (b − x)ε for all x ∈ I. Therefore, for (2.8) we obtain ⏐∫ ⏐ qx |y(x) − u(x)| = e lim ⏐⏐ s→b
≤
s
x
e
−µt
(
⏐ ∫ ) ⏐ z(t) − g(t) dt⏐⏐ ≤ lim s→b
(b − x)2 (b − a)2 ε ≤ ε, 2 2
s
ε(b − t)dt
x
for all x ∈ I.
□
In next Remark, we consider the cases that one of real parts for the roots of the characteristic equation is a negative number. In these cases, we use little different functions for z(x) and u(x).
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Remark 2.3. Case 1, p < 0 and q < 0: In this case, we define the functions, g(x) and z(x) as ( ) ∫ x g(x) = y ′ (x) − µy(x), z(x) = lim g(s)eλ(x−s) + eλx r(t)e−λt dt s→a
s
for all x ∈ I. Then, similar to (2.3), we get z ′ (x) = λz(x) + r(x) and in view of (2.2), the inequality (2.4) is also true for all x ∈ I. Moreover, similar to (2.5), we have ⏐ ⏐∫ x ⏐ ( ) ⏐ e−λt r(t) − g ′ (t) + λg(t) dt⏐⏐ |z(x) − g(x)| = epx lim ⏐⏐ s→a ∫ sx ⏐ ⏐ px ≤ e lim e−pt ⏐g ′ (t) − λg(t) − r(t)⏐dt s→a s ) ε( ε ≤− 1 − ep(x−a) ≤ − , for all x ∈ I. p p Next, for the function u(x) in the proof of Theorem 2.1, we define ( ) ∫ x u(x) = lim y(s)eµ(x−s) + eµx z(t)e−µt dt , s→a
for all x ∈ I.
s
Then, by following the process of the proof for Theorem 2.1, one notes that u is a solution to (1.4) and satisfies |y(x) − u(x)| ≤
ε , pq
for all x ∈ I.
Case 2, p < 0 and q > 0: We define the functions, g(x) and z(x) as Case 1 to get |z(x) − g(x)| ≤ −
ε p
for all x ∈ I. And, for the function u(x) in the proof of Theorem 2.1, we define ( ) ∫ s −µ(s−x) µx −µt u(x) = lim y(s)e −e z(t)e dt , for all x ∈ I. s→b
x
Then u is a solution to (1.4) and satisfies |y(x) − u(x)| ≤ −
ε . pq
Case 3, p < 0 and q = 0: In this case, we define the functions, g(x) and z(x) as Case 1 to get |z(x) − g(x)| ≤ −
ε p
for all x ∈ I. And, for the function u(x) in the proof of Theorem 2.1, we define ( ) ∫ x µ(x−s) µx −µt u(x) = lim y(s)e +e z(t)e dt , for all x ∈ I. s→a
s
Then u is a solution to (1.4) and satisfies ε |y(x) − u(x)| ≤ − (b − a), p
for all x ∈ I.
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Remark 2.4. In [15], Li and Shen also obtained the interesting results about the Hyers–Ulam stability of the second-order linear differential equations. Main difference is that they considered the real constant coefficient linear differential equations while we considered the complex constant coefficient linear differential equations. Also, they restricted to have two different positive real roots p and q for the characteristic equations of the linear differential equations. But, in this paper we do not have any restrictions for the roots of the characteristic equations. As a result we can apply our result to the time independent Schr¨ odinger equations. Moreover, they restricted for the bounded domain I = (a, b) while we extended to the unbounded domain −∞ ≤ a < b ≤ ∞ in Theorem 2.1. And with their assumptions we obtained same estimations with that in their paper.
3. Applications We consider the Schr¨ odinger equation for one spatial dimension −
ℏ2 d2 ψ(x) + V ψ(x) = Eψ(x), 2m dx2
(3.1)
where the wave function ψ : (a, b) → C is twice continuously differentiable, V is the potential energy, and E is the total energy of the quantum system. In particular, we assume that V ≤ E. The roots of the characteristic equation of Schr¨ odinger equation (3.1) are √ √ 2m(E − V ) 2m(E − V ) λ = −i and µ = i . ℏ2 ℏ2 Then we know that p = ℜ(λ) = 0 and q = ℜ(µ) = 0. Hence, by Theorem 2.2, if the function ψ satisfies the inequality ⏐ ⏐ ⏐ ℏ2 d2 ψ(x) ⏐ ⏐− ⏐ ≤ ε, for all x ∈ I, (3.2) + V ψ(x) − Eψ(x) ⏐ 2m dx2 ⏐ then there exists a solution ϕ : I → C of the differential equation (3.1) such that |ψ(x) − ϕ(x)| ≤
mε (b − a)2 , ℏ2
for all x ∈ I.
Acknowledgments This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2015R1D1A1A02061826) and (NRF2015R1D1A1A01059467). This work was also supported by 2016 Hongik University Research Fund and Hallym University Research Fund (HRF-201608-004). References
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