Quantum Chaos Of 2D Pseudo-integrable Systems

In this work,we study quantum chaos of a class of pseudo-integrable systems systematically,pseudo-integrable systems is a system with each angle of the polygon is rational multiples of ?,we studied one class of right triangle with each angle is a rational multiples of ?.In other words,we study the spectral statistics and eigenfunctions of Dirac equation and Schrodinger equation of pseudo-integrable systems.By using boundary integral methods,we can obtain the eigenvalues and eigenfunctions of Dirac equation and Schrodinger equation in any energy level regions.We can obtain eigenvalues and eigenfunctions of Schrodinger equations and Dirac equations,the rate of loss eigenvalues of GOE(Gaussian orthogonal ensemble)system is less than 1% and the rate of loss eigenvalues of GUE(Gaussian unitary ensemble)system less than 0.5%.Also,we develop conformal mapping methods which can solve Schrodinger equations and Dirac equations of arbitrarily polygon system.In classical mechanics,the classical invariant surfaces of particles moving in a two dimension system equivalent to a multi-handled sphere,the number of handled is called genus,the move directions of particle in classical mechanics increase with the increase of genus,so the instability of system increase.For quantum system,the spectral statistics change from Poisson distribution to Winger distribution with the increase of genus.This means the spectral statistics of Schrodinger equation change from Poisson distribution to GOE distribution,the spectral statistics of Dirac equation change from Poisson distribution to GUE distribution.For pseudo-integrable of non-relativistic quantum system,researcher have found the the spectral statistics depend on genus and energy level regions,with the increase of genus,the spectral statistics change from Poisson distribution to GOE distribution.If the system only a small change from integrable system,in the low energy level regions,the spectral statistics is near to Poisson distribution,for the high level regions,the spectral statistics is near to GOE distribution.Our discussions agree with those results in some degree.For small genus non-relativistic quantum system,we found for right triangle with genus change from 2 to 6,the spectral statistics is change from Poisson distribution to GOE distribution in general,but for the system with genus equal to 5,the average spectral statistics is deviate this trend.For the triangles with genus equal to33,we found the spectral statistics of those systems in low energy level regions depend on the degree of those systems deviate from the integrable system,near to integrable system,the spectral statistics are near to Poisson distribution,far from the integrable system,the spectral statistics are near to GOE distribution.Also,we studied the spectral statistics of near integrable(those systems are very close to integrable system)non-relativistic quantum system.Because of those systems have large genus,so as the previous researcher think,the spectral statistics should be GOE,at least for semi-classical regions.We found in the low energy level regions the spectral statistics is near to Poisson distribution,with the increase of energy level the spectral statistics move to GOE distribution,but for the energy levels around the 512000-th,the spectral statistics still locate in the middle of Poisson distribution to GOE distribution.For pseudo-integrable of relativistic quantum system,we studied the right triangle with genus change from 2 to 6,we found in the low energy level regions,the spectral statistics is change from GOE distribution to GUE distribution in general,but for the system with genus equal to 6,the average spectral statistics is deviate this trend.For the triangles with genus equal to 33,we found the spectral statistics of those systems in low energy level regions depend on the degree of those systems deviate from the integrable system,near to integrable system,the spectral statistics are near to GOE distribution,far from the integrable system,the spectral statistics are GUE distribution.