
The Eigenvalue problem for a region G in the plane asks to find
explicit formulas for the eigenvalues and eigenvectors of the linear
operator T=D_{xx} + D_{yy} on smooth functions in G
which vanish on the boundary of G.
For regions in which the billiard ball problem is explicitly
solvable, one expects the eigenvalues and eigenfunctions
of the linear operator T=D_{xx} + D_{yy} to be computable
explicitly also.
The PDE f_{t}(t,x,y) = i T f(t,xy,z) is called the
Schrödinger equation for a free particle in the region G
Knowing the complex valued function f(t,x,y) allows to compute
the probability that t>e particle is inside some region D by
integrating f^{2} over D and dividing this by
the integral of f^{2} over G. Because for eigenfunctions
T f = L f one can easily find the solution f(t,x,y) = exp(i L t) f(0,x,y),
the evolution of a general function f is obtained by writing f as a linear
combination of eigenfunctions and summing up the evolved eigenfunctions.
