# Determination of a Control Parameter for the Difference Schrödinger Equation

Authors: Allaberen Ashyralyev; Mesut Urun
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The first order of accuracy difference scheme for the numerical solution of the boundary value problem for the differential equation with parameterp, i(du(t)/dt)+Au(t)+iu(t)=f(t)+p, 0<t<T, u(0)=φ, u(T)=ψ, in a Hilbert space H with self-adjoint positive definite operator A is constructed. The well-posedness of this difference scheme is established. The stability inequalities for the solution of difference schemes for three different types of control parameter problems for the Schrödinger equation are obtained.


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## Body

## 1. Introduction: Difference Scheme

The theory and applications of well-posedness of inverse problems for partial differential equations have been studied extensively in a large cycle of papers (see, e.g., [1–24] and the references therein).Our goal in this paper is to investigate Schrödinger equations with parameter. In the paper [25], the boundary value problem for the differential equation with parameter p(1)idu(t)dt+Au(t)+iu(t)=f(t)+p,0<t<T,u(0)=φ,u(T)=ψ
in a Hilbert space H with self-adjoint positive definite operator A was studied. The well-posedness of this problem was established. The stability inequalities for the solution of three determinations of control parameter problems for the Schrödinger equation were obtained. In the present paper, the first order of accuracy Rothe difference scheme
(2)iτ-1(uk-uk-1)+Auk+iuk=φk+p,φk=f(tk),tk=kτ,1≤k≤N,Nτ=T,u0=φ,uN=ψ
for the approximate solution of the boundary value problem (1) for the differential equation with parameter p is presented. It is easy to see that
(3)uk=vk+(A+iI)-1p,p=(A+iI)(ψ-vN),
where {vk}k=0N is the solution of the following single-step difference scheme:
(4)iτ-1(vk-vk-1)+Avk+ivk=φk,φk=f(tk),tk=kτ,1≤k≤N,Nτ=T,v0-vN=φ-ψ.The theorem on well-posedness of difference problem (2) is proved. In practice, the stability inequalities for the solution of difference schemes for the approximate solution of three different types of control parameter problems are obtained.The paper is organized as follows. Section1 is the introduction. In Section 2, the main theorem on stability of difference problem (2) is established. In Section 3, theorems on the stability inequalities for the solution of difference schemes for the approximate solution of three different types of control parameter problems are obtained. In Section 4, numerical results are given. Finally, Section 5 is the conclusion.


## 2. The Main Theorem on Stability

In this section, we will study the stability of difference scheme (2).Let[0,T]τ={tk=kτ,k=1,…,N,Nτ=T} be the uniform grid space with step size τ>0, where N is a fixed positive integer. Throughout the present paper, ℱ([0,T]τ,H) denotes the linear space of grid functions φτ={φk}1N with values in the Hilbert space H. Let 𝒞τ(H)=𝒞([0,T]τ,H) be the Banach space of bounded grid functions with the norm
(5)∥φτ∥𝒞τ(H)=max1≤k≤N∥φk∥H.
Let us start with a lemma we need below. We denote that R=((1+τ)I-iτA)-1 is the step operator of problem (2).Lemma 1.
Assume thatA is a positive definite self-adjoint operator. The operator I-RN has an inverse Tτ=(I-RN)-1 and the following estimate is satisfied:
(6)∥Tτ∥H→H≤M(δ).Proof.
The proof of estimate (6) is based on the triangle inequality and the estimate
(7)∥(I-RN)-1∥H→H≤supδ≤μ11-|(1+τ(1-iμ))-N|≤11-((1+τ)2+(τδ)2)-N/2≤μ(δ).
Now, let us obtain the formula for the solution of problem (2). It is clear that the first order of accuracy difference scheme
(8)iτ-1(uk-uk-1)+Auk+iuk=p+φk,φk=f(tk),tk=kτ,1≤k≤N,Nτ=T,u0=φ
has a solution and the following formula
(9)uk=Rkφ-i∑j=1k‍Rk-j+1(p+φj)τ,1≤k≤N
is satisfied. Applying formula (9) and the boundary condition
(10)uN=ψ,
we can write
(11)ψ=RNφ-i∑j=1N‍RN-j+1φjτ-i∑j=1N‍RN-j+1τp.
Since
(12)-i∑j=1N‍RN-j+1τ=-i(I-iA)-1(I-R)∑j=1N‍RN-j=-i(I-iA)-1(I-RN),
we have that
(13)ψ=RNφ-i∑j=1N‍RN-j+1φjτ-i(I-iA)-1(I-RN)p.
By Lemma 1, we get
(14)p=Tτ(∑j=1N‍(I-iA)ψ-(I-iA)RNφ-∑j=1N‍(I-iA)RN-j+1φjτ).
Using (9) and (14), we get
(15)uk=Rkφ-i∑j=1k‍Rk-j+1φjτ+∑j=1k‍Rk-j+1τTτ(∑j=1N‍(I-iA)ψ-(I-iA)RNφ-∑j=1N‍(I-iA)RN-j+1φjτ),0001≤k≤N.
Since
(16)∑j=1k‍Rk-j+1τ=(I-iA)-1(I-R)∑j=1k‍Rk-j=(I-iA)-1(I-Rk),
we have that
(17)uk=Rkφ+∑j=1k‍Rk-j+1φjτ+(I-Rk)Tτ(ψ-RNφ-∑j=1N‍RN-j+1φjτ),0001≤k≤N.
Hence, difference scheme (2) is uniquely solvable and for the solution, formulas (14) and (17) hold.Theorem 2.
Suppose that the assumption of Lemma1 holds. Let φ,ψ∈D(A). Then, for the solution ({uk}k=1N,p) of difference scheme (2) in Cτ(H)×H, the estimates
(18)∥p∥H≤M[max2≤k≤N∥φk-φk-1τ∥H∥Aφ∥H+∥Aψ∥H+∥φ1∥H+max2≤k≤N∥φk-φk-1τ∥H],(19)∥{uk}k=1N∥Cτ(H)≤M[∥φ∥H+∥ψ∥H+∥{φk}k=1N∥Cτ(H)]
hold, where M is independent of τ, φ, ψ, and {φk}k=1N.Proof.
From formulas (9) and (14), it follows that
(20)p=Tτ[∑j=2N‍Aψ-RNAφ-φN+RNφ1-∑j=2N‍RN-j+1(φj-1-φj)].
Using this formula, the triangle inequality, and estimate (6), we obtain
(21)∥p∥H≤∥Tτ∥H→H(∑j=2N‍∥Aψ∥H+∥RN∥H→H∥Aφ∥H+∑j=2N‍∥RN-j+1∥H→H∥φj-φj-1∥H+∥φN∥+∥RN∥H→H∥φ1∥H∑j=2N‍)≤M[∥Aφ∥H+∥Aψ∥H+∥{φk}k=1N∥Cτ(1)(H)].
Estimate (18) is proved. Using formula (17), the triangle inequality, and estimate (6), we obtain
(22)∥uk∥H≤[∑j=1N‍∥RN-j+1∥H→H∥φj∥Hτ)∥Rk∥H→H∥φ∥H+∑j=1k‍∥Rk-j+1∥H→H∥φj∥Hτ+(1+∥Rk∥H→H)∥Tτ∥H→H×(∑j=1N‍∥ψ∥H+∥RN∥H→H∥φ∥H+∑j=1N‍∥RN-j+1∥H→H∥φj∥Hτ)]≤M[∥φ∥H+∥ψ∥H+∥{φk}k=1N∥Cτ(H)]
for any k. From that, it follows estimate (19). This completes the proof of Theorem 2.


## 3. Applications

Now, we consider the simple applications of main Theorem2.First, the boundary value problem for the Schrödinger equation(23)iut-(a(x)ux)x+δu+iu=p(x)+f(t,x),0<t<T,0<x<1,u(0,x)=φ(x),u(T,x)=ψ(x),0≤x≤1,u(t,0)=u(t,1),ux(t,0)=ux(t,1),0≤t≤T
is considered. Problem (23) has a unique smooth solution (u(t,x),p(x)) for the smooth functions a(x)≥a>0, x∈(0,1), δ>0, a(1)=a(0), φ(x), ψ(x) (x∈[0,1]), and f(t,x) (t∈(0,T), x∈(0,1)). This allows us to reduce the boundary value problem (23) to the boundary value problem (1) in a Hilbert space H=L2[0,1] with a self-adjoint positive definite operator Ax defined by formula
(24)Axu(x)=-(a(x)ux)x+δu
with domain
(25)D(Ax)={u(x):u(x),ux(x),(a(x)ux)x∈L2[0,1],u(1)=u(0),ux(1)=ux(0)}.
The discretization of problem (23) is carried out in two steps. In the first step, we define the grid space
(26)[0,1]h={x=xn:xn=nh,0≤n≤M,Mh=1}.
Let us introduce the Hilbert space L2h=L2([0,1]h) of the grid functions
(27)φh(x)={φn}1M-1
defined on [0,1]h, equipped with the norm
(28)∥φh∥L2h=(∑x∈[0,1]h‍|φ(x)|2h)1/2.
To the differential operator Ax defined by formula (24), we assign the difference operator Ahx by the formula
(29)Ahxφh(x)={-(a(x)φx¯)x,n+δφn}1M-1
acting in the space of grid functions φh(x)={φn}1M-1 satisfying the conditions φ0=φM, φ1-φ0=φM-φM-1. It is well known that Ahx is a self-adjoint positive definite operator in L2h. With the help of Ahx, we reach the boundary value problem
(30)iduh(t,x)dt+Ahxuh(t,x)+iuh(t,x)=ph(x)+fh(t,x),0<t<T,x∈[0,1]h,uh(0,x)=φh(x),uh(T,x)=ψh(x),x∈[0,1]h.In the second step, we replace (30) with the difference scheme (2)
(31)iukh(x)-uk-1h(x)τ+Ahxukh(x)+iukh(x)=ph(x)+fkh(x),fkh(x)=fh(tk,x),tk=kτ,Nτ=T,1≤k≤N,x∈[0,1]h,uh(0,x)=φh(x),uh(T,x)=ψh(x),x∈[0,1]h.Theorem 3.
The solution pairs({ukh(x)}0N,ph(x)) of problem (31) satisfy the stability estimates
(32)∥ph∥L2h≤M1[∥{fkh}1N∥Cτ(1)(L2h)∥φh∥L2h+∥Ahxφh∥L2h+∥ψh∥L2h+∥Ahxψh∥L2h+∥{fkh}1N∥Cτ(1)(L2h)],∥{ukh}1N∥Cτ(L2h)≤M2[∥φh∥L2h+∥ψh∥L2h+∥{fkh}1N∥Cτ(L2h)],
where M1 and M2 do not depend on φh, ψh, and fkh, 1≤k≤N. Here, Cτ(1)(L2h) is the grid space of grid functions {fkh}1N defined on [0,T]τ×[0,1]h with norm
(33)∥{fkh}1N∥Cτ(1)(L2h)=∥{fkh}1N∥Cτ(L2h)+sup1≤k<k+r≤N∥fk+rh-fkh∥L2hrτ,∥{fkh}1N∥Cτ(L2h)=max1≤k≤N∥fkh∥L2h.The proof of Theorem3 is based on formulas for ph(x) and {ukh(x)}1N and the symmetry property of operator Ahx.Second, letΩ=(x=(x1,…,xn):0<xk<1,k=1,…,n) be the unit open cube in the n-dimensional Euclidean space ℝn with boundary S, Ω¯=Ω∪S. In [0,T]×Ω, the boundary value problem for the multidimensional Schrödinger equation
(34)i∂u(t,x)∂t-∑r=1n‍(ar(x)uxr)xr+iu=p(x)+f(t,x),00000000000000x=(x1,…,xn)∈Ω,0<t<T,u(0,x)=φ(x),u(T,x)=ψ(x),x∈Ω¯,u(t,x)=0,x∈S,0≤t≤T
is considered. Here, ar(x)≥a>0 (x∈Ω), f(t,x) (t∈(0,T), x∈Ω), and φ(x), ψ(x) (x∈Ω¯) are given smooth functions.We consider the Hilbert spaceL2(Ω¯) of all square integrable functions defined on Ω¯, equipped with the norm
(35)∥f∥L2(Ω¯)=(∫‍⋯∫‍x∈Ω¯|f(x)|2dx1⋯dxn)1/2.
Problem (34) has a unique smooth solution (u(t,x),p(x)) for the smooth functions φ(x), ψ(x), ar(x), and f(t,x). This allows us to reduce the problem (34) to the boundary value problem (1) in the Hilbert space H=L2(Ω¯) with a self-adjoint positive definite operator Ax defined by the formula
(36)Axu(x)=-∑r=1n‍(ar(x)uxr)xr
with domain
(37)D(Ax)={u(x):u(x),uxr(x),(ar(x)uxr)xr∈L2(Ω¯),1≤r≤n,u(x)=0,x∈S(ar(x)uxr)xr∈L2(Ω¯)}.
The discretization of problem (34) is carried out in two steps. In the first step, we define the grid space
(38)Ω¯h={x=xr=(h1j1,…,hnjn),j=(j1,…,jn),0≤jr≤Nr,Nrhr=1,r=1,…,n},Ωh=Ω¯h∩Ω,Sh=Ω¯h∩S
and introduce the Hilbert space L2h=L2(Ω¯h) of the grid functions
(39)φh(x)={φ(h1j1,…,hnjn)}
defined on Ω¯h, equipped with the norm
(40)∥φh∥L2h=(∑x∈Ωh‍|φh(x)|2h1⋯hn)1/2.
To the differential operator Ax defined by formula (36), we assign the difference operator Ahx by the formula
(41)Ahxuh=-∑r=1n‍(αr(x)uxrh)xr,jr,
where Ahx is known as self-adjoint positive definite operator in L2h, acting in the space of grid functions uh(x) satisfying the conditions uh(x)=0 for all x∈Sh. With the help of the difference operator Ahx, we arrive to the following boundary value problem:
(42)iuth(t,x)+Ahxuh(t,x)+iuh(t,x)=ph(x)+fh(t,x),0<t<T,x∈Ωh,uh(0,x)=φh(x),uh(T,x)=ψh(x),x∈Ωh
for an infinite system of ordinary differential equations.The first order of accuracy difference scheme for the solution of problem (42) is
(43)iukh(x)-uk-1h(x)τ+Ahxukh(x)+iukh(x)=ph(x)+fkh(x),fkh(x)=fh(tk,x),tk=kτ,Nτ=T,1≤k≤N,x∈Ωh,uh(0,x)=φh(x),uh(T,x)=ψh(x),x∈Ωh.Theorem 4.
The solution pairs({ukh(x)}0N,ph(x)) of problem (43) satisfy the stability estimates
(44)∥ph∥L2h≤M1[∥{fkh}1N∥Cτ(1)(L2h)∥φh∥L2h+∥Ahxφh∥L2h+∥ψh∥L2h+∥Ahxψh∥L2h+∥{fkh}1N∥Cτ(1)(L2h)],∥{ukh}1N∥Cτ(L2h)≤M2[∥φh∥L2h+∥ψh∥L2h+∥{fkh}1N∥Cτ(L2h)],
where M1 and M2 do not depend on φh, ψh, and fkh, 1≤k≤N. Here, Cτ(1)(L2h) is the grid space of grid functions {fkh}1N defined on [0,T]τ×Ωh¯ with norm
(45)∥{fkh}1N∥Cτ(1)(L2h)=∥{fkh}1N∥Cτ(L2h)+sup1≤k<k+r≤N∥fk+rh-fkh∥L2hrτ,∥{fkh}1N∥Cτ(L2h)=max1≤k≤N∥fkh∥L2h.The proof of Theorem4 is based on Theorem 3 and the symmetry property of the operator Ahx is defined by formula (34) and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L2h.Theorem 5.
For the solutions of the elliptic difference problem [26]
(46)Ahxuh(x)=ωh(x),x∈Ωh,uh(x)=0,x∈Sh,
the following coercivity inequality holds:
(47)∑r=1n∥uxrxr-h∥L2h≤M∥ωh∥L2h,
where M does not depend on h and ωh.Third, in[0,T]×Ω, the boundary value problem for the multidimensional Schrödinger equation
(48)i∂u(t,x)∂t-∑r=1n‍(ar(x)uxr)xr+δu+iu=p(x)+f(t,x),000000000000000000x=(x1,…,xn)∈Ω,0<t<T,u(0,x)=φ(x),u(T,x)=ψ(x),x∈Ω¯,∂u(t,x)∂n→=0,x∈S,0≤t≤T
with the Neumann condition is considered. Here, n→ is the normal vector to S, δ>0, and ar(x)≥a>0(x∈Ω), f(t,x) (t∈(0,T), x∈Ω), and φ(x), ψ(x) (x∈Ω¯) are given smooth functions.Problem (48) has a unique smooth solution (u(t,x),p(x)) for the smooth functions φ(x), ψ(x), ar(x), and f(t,x). This allows us to reduce the problem (48) to the boundary value problem (1) in the Hilbert space H=L2(Ω¯) with a self-adjoint positive definite operator Ax defined by formula
(49)Axu(x)=-∑r=1n‍(ar(x)uxr)xr+δu
with domain
(50)D(Ax)={∂u(x)∂n→u(x):u(x),uxr(x),(ar(x)uxr)xr∈L2(Ω¯),1≤r≤n,∂u(x)∂n→=0,x∈S}.
The discretization of problem (48) is carried out in two steps. In the first step, we define the difference operator Ahx by the formula
(51)Ahxuh=-∑r=1n‍(αr(x)uxrh)xr,jr+δuh,
where Ahx is known as self-adjoint positive definite operator in L2h, acting in the space of grid functions uh(x) satisfying the conditions Dhuh(x)=0 for all x∈Sh. Here, Dh is the approximation of the operator ∂·/∂n→. With the help of the difference operator Ahx, we arrive to the following boundary value problem:
(52)iuth(t,x)+Ahxuh(t,x)+iuh(t,x)=ph(x)+fh(t,x),0<t<T,x∈Ωh,uh(0,x)=φh(x),uh(T,x)=ψh(x),x∈Ωh
for an infinite system of ordinary differential equations.The first order of accuracy difference scheme for the solution of problem (52) is
(53)iukh(x)-uk-1h(x)τ+Ahxukh(x)+iukh(x)=ph(x)+fkh(x),fkh(x)=fh(tk,x),tk=kτ,Nτ=T,1≤k≤N,x∈Ωh,uh(0,x)=φh(x),uh(T,x)=ψh(x),x∈Ωh.Theorem 6.
The solution pairs({ukh(x)}0N,ph(x)) of problem (53) satisfy the stability estimates
(54)∥ph∥L2h≤M1[∥{fkh}1N∥Cτ(1)(L2h)∥φh∥L2h+∥Ahxφh∥L2h+∥ψh∥L2h+∥Ahxψh∥L2h+∥{fkh}1N∥Cτ(1)(L2h)],∥{ukh}1N∥Cτ(L2h)≤M2[∥φh∥L2h+∥ψh∥L2h+∥{fkh}1N∥Cτ(L2h)],
where M1 and M2 do not depend on φh, ψh, and fkh,1≤k≤N.The proof of Theorem6 is based on Theorem 2 and the symmetry property of the operator Ahx is defined by formula (51) and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L2h.Theorem 7.
For the solution of the elliptic difference problem [26]
(55)Ahxuh(x)=ωh(x),x∈Ωh,Dhuh(x)=0,x∈Sh,
the following coercivity inequality holds:
(56)∑r=1n∥uxrxr-h∥L2h≤M∥ωh∥L2h,
where M does not depend on h and ωh.


## 4. Numerical Results

In present section, for numerical analysis, the following boundary value problem(57)i∂u(t,x)∂t-∂2u(t,x)∂x2+iu(t,x)=p(x)+f(t,x),x∈(0,π),t∈(0,1),u(0,x)=sinx,u(1,x)=e-1sinx,x∈[0,π],u(t,0)=u(t,π)=0,t∈[0,1]
is considered. The exact solution of problem (57) is u(t,x)=e-tsinx and p(x)=sinx.The first order of accuracy difference scheme(58)iunk-unk-1τ-un+1k-2unk+un-1kh2+iunk=φnk+p(xn),1≤k≤N,1≤n≤M-1,φnk=f(tk,xn)=(e-tk-1)sinxn,tk=kτ,0≤k≤N,Nτ=1,xn=nh,1≤n≤M-1,Mh=π,un0=sin(xn),unN=e-1sin(xn),xn=nh,0≤n≤M,u0k=uMk=0,0≤k≤N
for the numerical solution of problem (57) is constructed.For obtaining the values ofp(xn) at the grid points, we will use the following equation:
(59)p(xn)=-e-1sin(xn+1)-2sin(xn)+sin(xn-1)h2+ie-1sin(xn)+vn+1N-2vnN+vn-1Nh2-ivnN,xn=nh,1≤n≤M-1,
where vsk, s=n±1, and n is the solution of the first order of accuracy difference scheme
(60)ivnk-vnk-1τ-vn+1k-2vnk+vn-1kh2+ivnk=φnk,φnk=f(tk,xn),tk=kτ,1≤k≤N,Nτ=1,xn=nh,1≤n≤M-1,Mh=π,vnN-vn0=(e-1-1)sin(xn),xn=nh,0≤n≤M,v0k=vMk=0,0≤k≤N
generated by difference scheme (58).Using the difference scheme (60), we obtain (N+1)×(M+1) system of linear equations and we can write them in the matrix form as
(61)Avn+1+Bvn+Cvn-1=Rφn,1≤n≤M-1,v0=vM=0~,
where
(62)C=A=[000·00000x0·000000x·0000········000·00x0000·000x](N+1)×(N+1),B=[-100·001yz0·0000yz·000·······000·yz0000·0yz](N+1)×(N+1).
Here,
(63)x=-1h2,y=-iτ,z=iτ+2h2+i,vs=[vs0⋮vsN](N+1)×1fors=n+1,n,n-1,φn=[(e-1-1)sinxnφn1⋮φnN-1φnN](N+1)×1.
So, we have the second-order difference equation with respect to n with matrix coefficients. Using the modified Gauss elimination method, we can obtain vnk, 0≤k≤N, 0≤n≤M.For the solution of the matrix equations, we seek the solution of the form(64)vn=αn+1vn+1+βn+1,n=M-1,…,2,1,vM=0~,
where αj and βj, j=1,…,M, are calculated as
(65)αn+1=-(B+Cαn)-1(A),βn+1=(B+Cαn)-1(Dφn-Cβn),
where α1 is (N+1)×(N+1) and β1 is (N+1)×1 zero matrix.Then, using (59), values of p(xn) at grid points are obtained. Replacing p(xn) in (58), we get (N+1)×(M+1) system of linear equations and it can be written in the matrix form
(66)A2un+1+B2un+C2un-1=Rθn,1≤n≤M-1,1u0=uM=0~,
where
(67)C2=C,A2=A,B2=[100·000yz0·0000yz·000·······000·yz0000·0yz](N+1)×(N+1).
Here,
(68)y=-iτ,z=iτ+2h2+i,us=[us0⋮usN](N+1)×1fors=n+1,n,n-1,θn=[sinxnφn1+p(xn)⋮φnN-1+p(xn)φnN+p(xn)](N+1)×1.
Using the modified Gauss elimination method again, we can obtain unk, 0≤k≤N, 0≤n≤M.We will give the results of the numerical analysis. The numerical solutions are recorded for different values ofN and M and unk represents the numerical solutions of the difference scheme at (tk,xn). Table 1 is constructed for N=M=20, 40, and 80, respectively and the errors are computed by the following formula:
(69)E=max1≤k≤N{∑n=1M‍|u(tk,xn)-unk|2h}1/2.
For their comparison, Table 2 is constructed when errors are computed by
(70)E=max1≤k≤N1≤n≤M|u(tk,xn)-unk|.
Table 3 is constructed for the error of p(x) at the nodes in maximum norm.Table 1
Error analysis for the exact solutionu(t,x).
Method
N
=
M
=
20
N
=
M
=
40
N
=
M
=
80
1st order of accuracy d.s.
0.0024
0.0012
6.0463 × 10−4Table 2
Error analysis for the exact solutionu(t,x).
Method
N
=
M
=
20
N
=
M
=
40
N
=
M
=
80
1st order of accuracy d.s.
0.0019
9.5692 × 10−4
4.8241 × 10−4Table 3
Error analysis forp(x).
Method
N
=
20
N
=
40
N
=
80
1st order of accuracy d.s.
0.0145
0.0072
0.0036


## 5. Conclusion

In the present study, the well-posedness of difference problem for the approximate solution of determination of a control parameter for the Schrödinger equation is established. In practice, the stability inequalities for the solution of difference schemes of the approximate solution of three different types of control parameter problems are obtained. The well-posedness of the boundary value problem (1) is established. The stability inequalities for the solution of difference schemes for three different types of control parameter problems for the Schrödinger equation are obtained. Moreover, applying the result of the monograph [15], the high order of accuracy single-step difference schemes for the numerical solution of the boundary value problem (1) can be presented. Of course, the stability inequalities for the solution of these difference schemes have been established without any assumptions about the grid steps.



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*Source: 548201-2013-11-11.xml*