Maybe it is a syntax problem, since I haven't been using Mathematica for much time, but I haven't been able to find a similar example in internet.

I am just trying to solve analitically the Sturm-Liouville problem of Schrödinger equation for a particle in a 2D box $[0,1 ] \times [0,1]$. Id est:

$ - \frac{\hbar^2}{2m} \Delta \psi(x,y) = E \psi(x, y) $

$\psi(0,y) = \psi(1,y) = \psi(x,0) = \psi(x,1) = 0$

This is what I wrote:

eqn = 2*pi^2*z[x, y] + (D[z[x, y], {x, 2}] + D[z[x, y], {y, 2}]) == 0;
bc = {z[x, 0] == 0, z[x, 1] == 0, z[0, y] == 0, z[1, y] == 0};
sol = DSolve[{eqn, bc}, z[x, y], {x, y}]

And this is what I get: