A standard energy eigenvalue problem for directly solving the stationary states of quantum billiards via boundary integral analysis

Abstract

By using the series expansions of the Bessel functions for the real and imaginary parts of the free particle Green’s function for the two-dimensional stationary states of quantum billiards, it can be shown that some components of the Green’s function are redundant which can be eliminated to make the boundary integral equation for the wave function of free particles inside quantum billiards independent of the wave numbers. This development leads to a much faster search for the energy eigenvalues of quantum billiards by scanning their wave numbers or the formulation of the standard eigenvalue problem which can directly be solved for the energy eigenvalues. Some numerical examples were used to demonstrate that the proposed technique is accurate, computationally effective and straightforward to be applied in practice.

Keywords

• Quantum billiards
• Energy eigenvalues
• Free-particle Green’s function
• Boundary integral method