Numerical aspects of eigenvalue and eigenfunction computations for chaotic quantum systems

Abstract

We give an introduction to some of the numerical aspects in quantum chaos. The classical dynamics of two--dimensional area--preserving maps on the torus is illustrated using the standard map and a perturbed cat map. The quantization of area--preserving maps given by their generating function is discussed and for the computation of the eigenvalues a computer program in Python is presented. We illustrate the eigenvalue distribution for two types of perturbed cat maps, one leading to COE and the other to CUE statistics. For the eigenfunctions of quantum maps we study the distribution of the eigenvectors and compare them with the corresponding random matrix distributions. The Husimi representation allows for a direct comparison of the localization of the eigenstates in phase space with the corresponding classical structures. Examples for a perturbed cat map and the standard map with different parameters are shown. Billiard systems and the corresponding quantum billiards are another important class of systems (which are also relevant to applications, for example in mesoscopic physics). We provide a detailed exposition of the boundary integral method, which is one important method to determine the eigenvalues and eigenfunctions of the Helmholtz equation. We discuss several methods to determine the eigenvalues from the Fredholm equation and illustrate them for the stadium billiard. The occurrence of spurious solutions is discussed in detail and illustrated for the circular billiard, the stadium billiard, and the annular sector billiard. We emphasize the role of the normal derivative function to compute the normalization of eigenfunctions, momentum representations or autocorrelation functions in a very efficient and direct way. Some examples for these quantities are given and discussed.

50 pages, 29 postscript figures. Some of the figures are in reduced quality only, see http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp02-4.html for better versions. On http://www.physik.uni-ulm.de/theo/qc/baec/qmaps.html the full Python programs to compute eigenvalues of quantum maps and spacings distributions can be found

References in corpus (4)

• Quantitative study of scars in the boundary section of the stadium billiard
• Autocorrelation function of eigenstates in chaotic and mixed systems
• Uniform approximation for diffractive contributions to the trace formula in billiard systems
• Chaotic eigenfunctions in momentum space

Cited by in corpus (7)

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• Records in the classical and quantum standard map
• Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards
• Eigenfrequencies of the randomly pinned drum and conductivity of graphene
• Spectral Geometry and the Semiclassical Approach
• Crossover from Diffusive to Ballistic Transport in Periodic Quantum Maps