SCHRÖDINGER EQUATION VIA WKB

It is interesting to study the Schrödinger equation. Therefore, the vibration of Schrödinger equation has been studied extensively and continues to receive attention in the current literature. Schrödinger equation can be found in [3, 5]. For Schrödinger problem with a uniform small parameter, one can use the method (WKB) Wentzel Kramers Brillouin, also known in the literature as the approximation Liouville Green. In [1, 2], WKB method was used for finding asymptotics at high frequency, this method is to obtain asymptotic series for solutions with a small parameter.


1.INTRODUCTION
It is interesting to study the Schrödinger equation. Therefore, the vibration of Schrödinger equation has been studied extensively and continues to receive attention in the current literature. Schrödinger equation can be found in [3,5]. For Schrödinger problem with a uniform small parameter, one can use the method (WKB) Wentzel -Kramers -Brillouin, also known in the literature as the approximation Liouville -Green. In [1,2], WKB method was used for finding asymptotics at high frequency, this method is to obtain asymptotic series for solutions with a small parameter.

MATHEMATICAL FORMULATION
In this section we find the approximate solution as a linear combination of two linearly independent solutions. Substituting solution in boundary conditions, a homogeous system of two equations is obtained. This system has non-trivial solutions when the determinant is zero.

SCHRÖDINGER EQUATION
In [4], it is obtained Schrödinger equation (1) where y is the wave function and It is assumed that Looking for the solution of the Schrödinger equation (1) in the form (2) where E is the energy. We obtain the equation The boundary value problem is (4)
In this section we state and solve Schrödinger equation at high frequency. We are searching for non-trivial solutions of problem (5)-(6). The main result of this paper is as follows Proof. Given that then equation (5) can be transformed into Following the traditional WKB method, the analytical solution can be replaced by a power series given by the following with  (x) and A j (x), j = 0,1,2,… are smooth functions and unknown.
Replacing (8) and each of the derivatives of v(x) in (7), we have the following expression (10)

Bulletin of Mathematical Sciences and Applications Vol. 6
Replacing (9) on both sides of (10) we obtain (11) Equating the coefficients of the asymptotic series in  and taking corresponding to  0 in (11) and using that A 0  0 as seen in the equation (15), it can be obtained (12) From equation (12) and choosing the corresponding equality to  1 in (11), we obtain (13) By equating the asymptotic series, more equations are obtained. We consider only the first two ones, because other equations are of order O( 2 ) . Since equation (12) has two real roots withopposite signs, we obtain (14) From the equation (13) and separating the functions x and integrating on both sides, it follows that (15) where C is a non-zero arbitrary constant. Therefore, differentiating with respect to x in the equation (14) and substituting (15), function A 0 (x) can be expressed as follows (16) Therefore, replacing (14) and (16) where c 1 ,c 2 are constants. It is noted that are linearly independent.

BMSA Volume 6
From the solution (19) and boundary conditions (6) yield a homogeneous system of two equations for two constants This system has nontrivial solutions when The equation (21)