Electronic states in a cylindrical quantum lens: Quantum chaos for decreasing system symmetry - art. no. 056237

The eigenvalue problem in a cylindrical lens geometry is studied. Using a c onformal mapping method, the shape of the boundary and the Hamiltonian for a free particle are reduced to those of a two-dimensional problem with circ ular symmetry. The wave functions are separated into two independent Hilber t subspaces due to the inherent symmetry of the problem. For small geometry deformations, the solutions are found by a specially designed perturbation approach, Comparisons between exact and perturbative solutions are made fo r different lens parameters. As the symmetry of the lens is reduced, the ch aracteristics of the spectrum and the corresponding spatial properties of t he wave functions are studied. Our results provide a family of billiard geo metries in which the electronic level spectrum is well characterized. In an alyzing the level spacing distribution of the spectrum. a strong deviation from the Poisson and Wigner limiting distributions is found as the boundary geometry changes. This intermediate distribution is indicative of a mixed phase space. also revealed explicitly in the classical Poincare maps we pre sent.