## 1 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.

As known, Schrödinger equation is a partial differential equation, that is, its solution is a function, not a number. This means that some partial differential equations can be solved, some can’t, and some can only be estimated. The exact and numerical solutions of the various partial differential equations have been investigated by using the different methods in works [1, 2, 3, 4, 5], [7, 8, 9, 10, 11], [13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. In this study, we analyze a Schrödinger equation whose solution is an estimate of particle. Many researchers have been studied the approximate and exact solutions of Schrödinger equation by using different methodologies such as Adomian Decomposition Method (ADM) [10, 16], Homotopy Perturbation Method (HPM) [3, 5, 14], Homotopy Analysis Method (HAM) [2, 4], Variational Iteration Method (VIM) [7, 18], Galerkin’s Method [1, 8, 9, 11, 13, 17, 19, 20, 21]. Besides, there is various solution techniques for Schrödinger equation.

In this paper, we regard a first type boundary value problem for linear Schrödinger equation in the form:

*ψ*=

*ψ*(

*x*,

*t*) is a wave function,

*l*and

*T*are positive numbers. Here, we will use the notations:

*x*∈

*I*= (0,

*l*),

*t*∈ [0,

*T*], Ω

_{t}= (0,

*l*) × (0,

*t*), Ω= Ω

_{T},

*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

We investigate the solutions of the boundary value problem (1.1) –(1.3) (BVP) under conditions (1.4)–(1.7). 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].

## 2 The Existence and Uniqueness of Solutions

In this section, we define the solution of BVP and prove a theorem which states the existence and uniqueness of the solution of BVP.

*A function**is said to be a solution of BVP, if it holds the conditions (1.1)–(1.3) for a.a* (*x*, *t*) ∈ Ω, *where**is a Sobolev space of all elements in L*_{2}(Ω) *having generalized derivatives up to order 2 and 1 with respect to variables x and t, respectiely, inclusive in L*_{2}(Ω). *Also*, *is a subspace of *

*Let the conditions (1.4)–(1.7) be satisfied. Then, BVP has a unique solution**which holds the estimate*

*where c*

_{0}

*is a positive constant independent from ϕ and f.*

We will prove the theorem (2.1) by the Galerkin’s method. By this method, the approximate solutions are searched in the form:

*u*

_{k}=

*u*

_{k}(

*x*) for

*k*= 1,2,.. generate a fundamental system in the space

*λ*of the problem:

_{k}*u*

_{k}=

*u*

_{k}(

*x*) corresponding to the eigenvalues

*λ*are real and ortogonal in the spaces

_{k}*L*

_{2}(

*I*),

*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*). Also, in (2.2), the coefficients

*k*= 1,2,..,

*N*, where

*ϕ*

_{k}→

*ϕ*strongly in

*T*].

We assert that the problem (2.4)–(2.5) has global solution on [0, *T* ]. To prove it, we give the next lemma:

*The coefficients**provide the estimate*

*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 (2.1) that all possible solutions of the Cauchy problem (2.4)–(2.5) are uniformly bounded on [0, *T* ], which implies that problem (2.4)–(2.5) has one global solution on [0, *T* ].

Let’s define a family of functions *l*_{N,k}(*t*) for *k,N* = 1,2,... such that *l*_{N,k}(*t*) = *ψ*^{N}(.,*t*),*u*_{k})_{L2(I)}. From the estimate (2.6), it is seen that the functions *l*_{N,k}(*t*) for *k,N* = 1,2,... are uniformly bounded on [0, *T* ]. So, the inequality

*c*

_{2}is independent from

*N,k*.

Now, let’s show that the functions *l*_{N,k}(*t*) are equicontinuous for fixed *k* and *N* ≥ *k*, *N*, *k* = 1,2,... on the interval [0, *T* ]. For this purpose, we integrate the *k* − *th* equation in (2.4) on (*t*, *t* + Δ*t*) and so, we get

*t*to all term at right-hand side of (2.8), we obtain

*k,N*= 1,2,..., where the positive constant

*c*

_{3}does not depend on

*N,k*,Δ

*t*. It follows from (2.9), that the functions

*l*

_{N,k}(

*t*) are equicontinuous on [0,

*T*].

Thus, from Ascoli-Arzela’s theorem [6], we can extract the subsequence {*l*_{Nm,k}(*t*)} from sequences {*l*_{N,k}(*t*)} for fixed *k* and *m* = 1,2,.. such that

*ψ*

^{Nm}} 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

*ɛ*such that |(

*ψ*

^{Nm}(

*x*,

*t*) −

*ψ*(

*x*,

*t*),

*g*)

_{L2(I)}|<

*ɛ*for all

*t*∈ [0,

*T*], ∀

*g*∈

*L*

_{2}(

*I*). 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*∈

*L*

_{2}(

*I*),

*s*. Also, since

*l*

_{Nm,k}(

*t*) →

*l*

_{k}(

*t*) as

*N*

_{m}→ ∞ it is written that for any

*ɛ*> 0

*s*, where

*c*

_{3}> 0 is independent from

*N*

_{m}. Similarly, it is clear that

*c*

_{4}> 0 is independent from

*N*

_{m}. Since the series

*g*∈

*L*

_{2}(

*I*), if we regard the converging of the series

*ɛ*> 0. Thus, consideringly this inequality if we use (2.13) and (2.14) in (2.12), we achieve

*N*

_{m}→ ∞ for ∀

*g*∈

*L*

_{2}(

*I*), ∀

*t*∈ [0,

*T*] and ∀

*ɛ*> 0, which follows that the sequence {

*ψ*

^{Nm}(

*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*^{Nm}
} is uniformly bounded from (2.6), we can extract a subsequence which weakly converges in *ψ*(*x*, *t*) defined by formula (2.11). For simplicity, let’s denote this subsequence as {*ψ*^{Nm}(*x*, *t*)}. That is, limit relations

*L*

_{2}(Ω), if we take the lower limit of estimate (2.6) for

*N*=

*N*

_{m}and as

*m*→ ∞ we have the inequalities

*ψ*(

*x*,

*t*) provides the estimate (2.1) and

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
*k* from 1 to *N′*≤ *N*_{m} and finally integrate over [0, *T* ]. Ultimately, we achieve the identity

*N′*≤

*N*. Thus, taking the limit of (2.19) for

_{m}*N*=

*N*

_{m}as

*m*→ ∞ and then using the limit relations (2.15)–(2.18), we obtain the identity

*N′*≤

*N*

_{m}. Since the set of functions

*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)

*ψ*(

*x*,

*t*) holds (1.1) for

*a.a*(

*x*,

*t*) ∈ Ω.

Similarly to the paper [20], we prove that the conditions (1.2) and (1.3) is fulfilled by limit function *ψ*(*x*, *t*). Thus, we arrive

Finally, let’s prove the uniquenes of solution of BVP in *ψ* and *ζ* in *ρ*(*x*, *t*) = *ψ*(*x*, *t*) − *ζ* (*x*, *t*).
Then, the function *ρ*(*x*, *t*) satisfies the following boundary value problem:

To obtain the uniquenes of the solution of BVP in _{t},
we have

*any t*∈ [0,

*T*]. So,

*ψ*(

*x*,

*t*) =

*ζ*(

*x*,

*t*) for

*any t*∈ [0,

*T*] and

*a.a x*∈

*I*. i.e., BVP has a unique solution in

## 3 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.

## References

- [1]↑
G. D. Akbaba, (2011), The optimal control problem with lions functional for the Schrödinger equation containing gradient with imaginary coefficient, Kafkas University, Inst. of Sci.&Tech., Master Thesis.

- [2]↑
A. K. Alomari, M. S. M. Noorani and R. Nazar, (2009), Explicit series solutions of some linear and nonlinear Schrödinger equations via the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, 14 (4), 1196–1207.

- [3]↑
J. Biazar and H. Ghazvini, (2007), Exact solutions for non-linear Schrödinger equations by He’s homotopy perturbation method, Physics Letters A., 366 (1–2), 79–84.

- [4]↑
B. Ghanbari, (2014), An analytical study for (2+1)- Dimensional Schrödinger Equation, The Scientific World Journal, http://dx.doi.org/10.1155/2014/438345.

- [5]↑
J. H. He, (2005), Application of homotopy perturbation method to nonlinear wave equation, Chaos, Solitons and Fractals, 26, 695–700.

- [6]↑
P. F. Hsieh and Y. Sibuya, (1999), Basic Theory of Ordinary Differential Equations, Springer Verlag, 468p., New York.

- [7]↑
Kh. Hosseinzadeh, (2017), An analytic approximation to the solution of Schrödinger equation by VIM, Applied Mathematical Sciences, 11(16), 813–818.

- [8]↑
A. D. Iskenderov and G.Y. Yagubov, (2007), Optimal control Problem with unbounded potential for multidimensional, nonlinear and nonstationary Schrödinger equation// Proceedings of the Lankaran State University, Natural Sciences series, pp. 3–56.

- [9]↑
A. D. Iskenderov, G. Y. Yagubov, M. A. Musayeva, (2012), Identification of Quantum Potentials, Baku, Casıoglu, 552 p.

- [10]↑
S. A. Khuri, (1998), A new approach to the cubic Schrödinger equation: an application of the decomposition technique, Appl. Math. and Comput., 97, 251–254.

- [11]↑
O. A. Ladyzhenskaya, (1985), The Boundary Value Problems of Mathematical Physics, Springer Verlag.

- [12]↑
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’ceva, (1968), Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. (English Trans.), Providence, RI.

- [13]↑
N. M. Mahmudov, (2007), Solvability of Boundary Value Problems for a Schrödinger Equation with Pure Imaginary Coefficient in the Nonlinear Part of This Equation, Proceedings of IMM of NAS of Azerbaijan, Vol.27, issue 35, pp.25–36.

- [14]↑
M. M. Mousa, S. F. Ragab and Z. Nturforsch, (2008), Application of the homotopy perturbation method to linear and nonlinear Schrödinger equation, Zeitschrift Fur Naturforschung A, 63 (3–4), 140–144.

- [15]↑
L.S. Pontryagin, (1962), Ordinary Differential Equtions, Addison-Wesley Publishing Comp.. (translated from the Russian)

- [16]↑
A. Sadighi and D.D. Ganji, (2008), Analytic treatment of linear and nonlinear Schrödinger equation: A study with homotopy perturbation and Adomian’s decomposition method, Pyhs. Lett. A, 372, 465–469.

- [17]↑
F. Toyoglu, (2012), Optimal control problems for the two-dimensional Schrödinger equation and their numerical solutions, Atatürk University, Inst. of Sci.&Tech., Ph. D. Thesis.

- [18]↑
A. M. Wazwaz, (2008), A study on linear and nonlinear Schrödinger Equation by variational iteration method, Chaos, Solitons and Fractals, 37, 1136–1142.

- [19]↑
G. Yagub, E. Aksoy, (2017), The Solvability of Initial Boundary Value Problem for Three Dimensional Nonlinear Schrödinger Equation With a Special Gradient Term, AIP Conference Proceedings, 1833, 020042.

- [20]↑
N. Yildirim Aksoy, Y. Kocak,Y. Ozeroglu, (2016), The Solvability of Initial Boundary Value Problems for Nonlinear Time-Dependent Schrödinger Equations, Quaestiones Mathematicae, 39 (6), 751–771.

- [21]↑
N. Yildirim Aksoy, (2017), Solution of a nonlinear Schrödinger equation with Galerkin’s Method, Igdır Univ. J. Inst. Sci. &Tech., 7(2), 225–239.

- [22]↑
A. Yokus, (2018), Numerical solution for space and time fractional order Burger type equation, Alexandria Engineering Journal, 57(3), 2085–2091.

- [23]↑
A. Yokus, (2018), Comparison of Caputo and Conformable derivatives for time-fractional Kortewegde Vries equation via the finite difference method, International Journal of Modern Physics B, 32(29) 1850365.

- [24]↑
A. Yokus and H. Bulut, (2019), On the numerical investigations to the Cahn-Allen equation by using finite difference method, An International Journal of Optimization and Control: Theories & Applications, 9(1) 18–23.

- [25]↑
A. Yokus, H. M. Baskonus, T. A. Sulaiman, H. Bulut, (2018), Numerical simulation and solutions of the two-component second order KdV evolutionary system, Numerical Methods for Partial Differential Equations, 34(1), 211–227.