Quantum mushroom billiards

Abstract

We report the first large-scale statistical study of very high-lying eigenmodes (quantum states) of the mushroom billiard proposed by L. Bunimovich in this journal, vol. 11, 802 (2001). The phase space of this mixed system is unusual in that it has a single regular region and a single chaotic region, and no KAM hierarchy. We verify Percival's conjecture to high accuracy (1.7%). We propose a model for dynamical tunneling and show that it predicts well the chaotic components of predominantly-regular modes. Our model explains our observed density of such superpositions dying as E^{-1/3} (E is the eigenvalue). We compare eigenvalue spacing distributions against Random Matrix Theory expectations, using 16000 odd modes (an order of magnitude more than any existing study). We outline new variants of mesh-free boundary collocation methods which enable us to achieve high accuracy and such high mode numbers orders of magnitude faster than with competing methods.

revised, corrected, expanded including new results on dynamical tunneling and level-spacing; 14 pages, 15 figures (4 new), resubmitted to CHAOS

References in corpus (11)

• Scarred eigenstates for quantum cat maps of minimal periods
• Stickiness in Hamiltonian systems: from sharply divided to hierarchical phase space
• Dynamical Tunneling in Mixed Systems
• Stickiness in mushroom billiards
• Poincaré Husimi representation of eigenstates in quantum billiards
• Spectral properties of Bunimovich mushroom billiards
• Weyl's law and quantum ergodicity for maps with divided phase space
• Isospectrality in Chaotic Billiards
• Behaviour of boundary functions for quantum billiards
• Numerical verification of Percival's conjecture in a quantum billiard
• Coincidence of length spectra does not imply isospectrality

Cited by in corpus (18)

• Geometrical structure of Laplacian eigenfunctions
• Dynamical tunneling in mushroom billiards
• Direct regular-to-chaotic tunneling rates using the fictitious integrable system approach
• Spectral fluctuations of billiards with mixed dynamics: from time series to superstatistics
• Quantum localization of chaotic eigenstates and the level spacing distribution
• Open Mushrooms: Stickiness revisited
• The intermediate level statistics in dynamically localized chaotic eigenstates
• Marginally Unstable Periodic Orbits in Semiclassical Mushroom Billiards
• Estimating Lyapunov exponents in billiards
• Few Islands Approximation of Hamiltonian System with divided Phase Space
• Magnetic field-induced control of transport in multiterminal focusing quantum billiards
• Coupling of bouncing-ball modes to the chaotic sea and their counting function
• Constrained quantum mechanics: chaos in non-planar billiards
• Metric deformation and boundary value problems in 2D
• Quantization of a Free Particle Interacting Linearly with a Harmonic Oscillator
• Casimir force between integrable and chaotic pistons
• Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues
• Metastability, chaos and spectrum tomography for Bose-Hubbard rings and chains