Suppose a domain with smooth boundary is given. It turns out that there is a discrete set of eigenfunctions which can be arranged in a sequence for which there is a nontrivial solution to the eigenfunction equation, and corresponding eigenfunctions. These eigenfunctions have some pleasant properties; for example, they can be chosen to form on orthonormal basis of the Hilbert space.
It turns out that asymptotic properties of the eigenfuctions is intimately related to the dynamical properties of the billiard flow. The billiard flow is a dynamical system whose state space is the set of unit tangent vectors. The flow at time t moves a unit tangent vector along the billiard trajectory starting at that tangent vector (straight line motion, bouncing of the boundary obeying the usual law of reflection) for distance t. For example, if the flow is chaotic then the eigenfunctions become equidistributed in the sense that the probability measures tend weakly to uniform measure.
There are many questions to consider.