The Solvability of First Type Boundary Value Problem for a Schrödinger Equation

The paper presents an first type boundary value problem for a Schrödinger equation. The aim of paper is to give the existence and uniqueness theorems of the boundary value problem using Galerkin’s method. Also, a priori estimate for its solution is given.


Introduction
The fundamental equation of quantum mechanics, Schrödinger eqution, is the basic non-relativistic wave equation which describes the behaviour of a single particle (on systems of particles) in a field of force. Its solution is called a wave function which give us information about the particle's behavior in time and space and the square of the wave function states the probability of finding the location of an particle in a given area. Schrödinger equation is roughly similar to Newton's equation. Schrödinger equation does for a quantum mechanical particle what Newton's Second Law does for a classical particle. When we solve Newton's equation we can find the position of a particle as depend on time. But, when we solve Schrödinger's equation we get a wave function stated the probability of finding the particle in some region in space varies as a function of time.
In this paper, we regard a first type boundary value problem for linear Schrödinger equation in the form: where ψ = ψ(x,t) is a wave function, l and T are positive numbers. Here, we will use the notations: a 0 > 0 is a given real number, the functions a 1 (x,t), a 2 (x), v(t) ∈ L 2 (0, T ) are the measurable real-valued which satisfy the conditions, respectively, ϕ(x) and f (x,t) are given complex-valued functions such that Here, the spacesW 2 2 (I),W 2,0 2 (Ω) are Sobolev space and are defined in [12] widely. We investigate the solutions of the boundary value problem (1. For this purpose, we shall apply a well known Galerkin's method to BVP. Also, we obtain an estimate for solutions of equation (1.1).
According to Galerkin's method, some fundamental system of linearly independent functions in the studied spaces is choosen and the approximate solutions by means of linearly independent functions are constituted. The solution of BVP is obtained as limits of approximate solutions calculated by this method [11].

1)
where c 0 is a positive constant independent from ϕ and f .
Proof. We will prove the theorem (2.1) by the Galerkin's method. By this method, the approximate solutions are searched in the form: where the functions u k = u k (x) for k = 1, 2, .. generate a fundamental system in the spaceW 2 2 (I) and are eigenfunctions corresponding to the eigenvalues λ k of the problem: This is a Sturm-Liouville problem. So, its eigenvalues are real and nonnegative and the eigenfunctions u k = u k (x) corresponding to the eigenvalues λ k are real and ortogonal in the spaces L 2 (I),W 1 2 (I),W 2 2 (I). Assume where d k are positive constants and let u k = u k (x) for k = 1, 2, .. be an orthonormal basis in the space L 2 (I).
for k = 1, 2, .., N, where ϕ k → ϕ strongly inW 2 2 (I). The system (2.4) is a system of first order linear nonhomogeneous ordinary differential equations with constant coefficients with respect to the unknowns C N k (t) and system (2.4) with (2.5) is a Cauchy problem. From [15], it is written that the problem (2.4)-(2.5) has locally at least one solution on [0, T ] .
We assert that the problem (2.4)-(2.5) has global solution on [0, T ] . To prove it, we give the next lemma: for N = 1, 2, .., where the positive constant c 1 does not depend on N.
The proof of lemma (2.1) is carried out as in [20]. We now turn to proof of the theorem (2.1). It follows from the Lemma After taking the absolute value of (2.7), applying the Cauchy-Schwarz inequality, we obtain and apply Cauchy-Schwarz inequality with respect to t to all term at right-hand side of (2.8), we obtain In above inequality, using the estimate (2.6), we get Thus, from Ascoli-Arzela's theorem [6], we can extract the subsequence {l N m ,k (t)} from sequences {l N,k (t)} for fixed k and m = 1, 2, .. such that and we claim that the subsequence ψ N m weakly converges to ψ(x,t) in L 2 (I), which this convergence are uniformly with respect to the variable t. That is, there is a positive number ε such that Since the space L 2 (I) is a separable Hilbert space, we can write any element g of L 2 (I) in the form g = . (2.12) Since g ∈ L 2 (I), as N m → ∞ it is written that for any ε > 0 for big enough values of s, where c 3 > 0 is independent from N m . Similarly, it is clear that where c 4 > 0 is independent from N m . Since the series is the rest of Fourier series of the function g ∈ L 2 (I), if we regard the converging of the series , we can write for any ε > 0. Thus, consideringly this inequality if we use (2.13) and (2.14) in (2.12), we achieve as N m → ∞ for ∀g ∈ L 2 (I), ∀t ∈ [0, T ] and ∀ε > 0, which follows that the sequence ψ N m (x,t) weakly converges to ψ(x,t) in L 2 (I) as uniformly with respect to t.
For N = N m , since the subsequence ψ N m is uniformly bounded from (2.6), we can extract a subsequence which weakly converges in W 2,1 2 (Ω) to ψ(x,t) defined by formula (2.11). For simplicity, let's denote this subsequence as ψ N m (x,t) . That is, limit relations are written.Thus, by using the limit relations (2.15)-(2.18) and the weakly lower semicontinuity of the norm on L 2 (Ω),if we take the lower limit of estimate (2.6) for N = N m and as m → ∞ we have the inequalities which is equivalent to ||ψ|| 2 (Ω) , it follows that the limit function ψ(x,t) provides the estimate (2.1) and ψ ∈ W 2,1 2 (Ω). Now, let's show that the function ψ(x,t) provides the equation (1.1) for a.a (x,t) ∈ Ω. After multiplying the k-th equation in (2.4) for N = N m with any continuous function η k (t) and let's sum the obtained equalities on k from 1 to N ≤ N m and finally integrate over [0, T ] .Ultimately,we achieve the identitŷ 19) where η N (x,t) = N ∑ k=1 η k (t)u k (x), N ≤ N m .Thus, taking the limit of (2.19) for N = N m as m → ∞ and then using the limit relations (2.15)-(2.18), we obtain the identitŷ Since the set of functions η N (x,t) are dense in L 2 (Ω), if we take the limit of above integral identity for N → ∞ we get the following identity for any η(x,t) ∈ L 2 (Ω) From (2.20), we can easily say that the limit function ψ(x,t) holds (1.1) for a.a (x,t) ∈ Ω.

Conclusion
In this paper, Galerkin's method have been succesfully applied to the linear Schrödinger equation with special gradient term. It was shown that the solution of BVP exists and it is unique. Also, an estimate satisfied by the solution function is obtained. Studied problem consists of a special gradient term and the coefficients of equation are more general than the former works. Especially, the coefficient a 1 depends on both the variables x and t. Because of the distinctness of considered equation with conditions, our problem differs from previous works in the literature.