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I am trying to solve an equation of the form as follows

$\left(\begin{array}{cc} -\frac{\hbar^{2}}{2m}\frac{\delta^{2}}{\delta z^{2}}+\sin^{2}\left(z\right) & z\\ z & \frac{\hbar^{2}}{2m}\frac{\delta^{2}}{\delta z^{2}}-\sin^{2}\left(z\right) \end{array}\right)\left(\begin{array}{c} u_{n}(z)\\ v_{n}(z) \end{array}\right)==\epsilon_{n}\left(\begin{array}{c} u_{n}(z)\\ v_{n}(z) \end{array}\right)$

Assuming required no. of boundary conditions but with no knowledge of the eigenvalues, please tell how to solve such a system of equations in Mathematica. Here $n$ is an index for the eigenfunction and the corresponding eigenvalue. Take $z$ as real and assume that $z$ runs on a finite lattice of fixed size of length $L=Na$ where $N$ is the number of sites on the lattice and a is the lattice constant. If you take $a==1$, take $N$ of the order of $10000$. There is a specified value of the eigenfunctions at the boundary of the lattice which you can take zero for the simplest case. Please multiply everything by units such that dimensions are consistent- Here the derivative term is the kinetic energy term while sin2z term is the potential term.

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Is this a Mathematica related question? If so do you have any Mathematica code. It would save the time to type the equation.... –  user21 Jun 20 '13 at 7:03
I want to know how to solve this in Mathematica. There are algorithms to solve the coupled eigenvalue differential equations but I have not been able to find a Mathematica command(s) that directly utilizes them for solving such a set of equations. And that or a combination of commands to help solve this, is what I am looking for. –  cleanplay Jun 20 '13 at 10:57
What kind of solution is expected? A closed form may be not attainable. Where seems to be no a ready procedure to find eigenvalues numerically, so probably an implementation of an algorithm you're mentioning would be needed. –  Andrew Jun 20 '13 at 11:41
You're units are dimensionally inconsistent. The derivative term has units of $s^{-1}$, but $\sin^2(z)$ is unitless. Also, what are the boundary conditions? They effect the eigenvalues of the system. Also, as this appears to be a Hamiltonian, is $z$ real? If not, shouldn't the matrix be Hermitian? –  rcollyer Jun 20 '13 at 13:24
@rcollyer I somtimes find in my own research that $2=1$ :D –  sebhofer Jun 20 '13 at 15:26
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