Quantum chaos in a ripple billiard.

We study the quantum chaos of a ripple billiard that has sinusoidal walls. We show that this type of ripple billiard has a Hamiltonian matrix that can be found exactly in terms of elementary functions. This feature greatly improves computation efficiency; a complete set of eigenstates from the ground state up to the 10,000th level can be calculated simultaneously. Nearest neighbor spacing of energy levels of a chaotic ripple billiard shows a Brody distribution (with a confidence level of 99% by chi(2) test) instead of the Gaussian orthogonal ensemble prediction. For high energy levels we observe scars and interesting patterns that have no resemblance to classical periodic orbits. Momentum localization of scarred eigenstates is also observed. We compare the scar associated localization with quantum dynamical Anderson localization by drawing the wave function distribution on basis state coefficients.