Chaotic quantum billiards on a torus.

[MathematicalPhysicist]

Hi, I want to do a similar statistics analysis as in the next paper:
http://www.phy.bris.ac.uk/people/Berry_mv/the_papers/Berry340.pdf

But the boundary conditions are on a two dimensioanl torus, so a solution will be of the form
$$u(R)=\sum_{j=1}^{\infty} \sum_{m,n=0}^{\infty} (A_{mn} sin(mXcos(\theta_j)+nYsin(\theta_j)+\phi_j) + B_{mn} cos(mXcos(\theta_j)+nYsin(\theta_j)+\phi_j)$$ (have I got it right?!) the problem arises when I impose on it a Dirchlet boundary condition (for example), cause I need f to satisfy: $$u(x,0)=u(x,2\pi)=0$$ and $$u(0,y)=u(2\pi,y)=0$$

Which doesn't look like something I can solve easily, can I?

Any help?

Thanks.

 

[jewbinson]

Surely there are familiar Fourier series tricks you can use with that summation expression...