Leaking Quantum Square Billiards

Open systems are often applied to model natural phenomena in many branches of physics and are commonly related to leaking systems. Problems with leaks also play an important role in the framework of dynamical theories. Quantum billiards are the quantum version of two-dimensional classical systems in which the particle dynamics is restricted by walls.
In this work, a closed and an open square quantum billiards are studied. The wave functions of the problems are obtained by the boundary wall method. Plane waves with different energies are scattered by the infinity potential defined on the billiard table boundary. The influence of the gap size {\it d} in a wall of the open system is studied in comparison with results of the closed version of the system. In this investigation the probability density function is computed for both the open and closed billiards.
Prelimenary results, show that the presence of a leak shifts the resonant states related to the closed billiard and large values of {\it d} can displace hight exited states. The flux of probability current is calculated on the leak and the relation between wave energy and the leaking was studied. In addition the energy level distribution is calculated and compared to Poisson and Wigner distributions.