How to apply boundary conditions when solving Schrodinger Eq

[ndchien]

Hi everyone,

I need to solve 1-D Schrodinger Eq. for an abitrary potential V(x), x = [x1,x2]. I have used the FEM, firstly is to discrete the distance [x1,x2] into N elements. Cubic interpolation polynomials (see L.R. Ram-Mohan, eq. (3.25), p. 72) were used to obtain the eigenvalue equation as follows:
H(i,j) . PSI(j)=E . U(i,j) . PSI(j)
where E is eigenvalue of energy, PSI(j) is the value of wavefunction (with odd value of j) at the N+1 odd nodes, while it is the value of the first derivative of the wavefunction (with even value of j) at the N+1 even nodes else, [j=1,2,..., (2*N+2)]. H(i,j) and U(i,j) are real symmetric banded matrices [ demension (2*N+2, 2*N+2)].
I have used the subroutine DSBGVX (in LAPACK library) to solve above eigenvalue equation. (all related subroutines for correctly running DSBGVX supplied by LAPACK Library also used).

The boundary conditions are:
PSI(x1)=PSI(x2)=0
It means that PSI at the first node and final node are eliminated as
PSI(1)=PSI(2*N+1)=0

How I can apply above boundary condition when I use DSBGVX to solve eigenvalue equation?

Acctually, I have try by deleting the columms 1 and N+1, rows 1 and N+1 of the matrices H and U (because PSI(1)=PSI(2*N+1)=0), and then solve eigenvalue eq. by calling DSBGVX.
However, the obtained results for the square well potential is incorrect!!!

What is my problem?

Thank you very much for any helps!