Small Disks and Semiclassical Resonances

Abstract

We study the effect on quantum spectra of the existence of small circular disks in a billiard system. In the limit where the disk radii vanish there is no effect, however this limit is approached very slowly so that even very small radii have comparatively large effects. We include diffractive orbits which scatter off the small disks in the periodic orbit expansion. This situation is formally similar to edge diffraction except that the disk radii introduce a length scale in the problem such that for wave lengths smaller than the order of the disk radius we recover the usual semi-classical approximation; however, for wave lengths larger than the order of the disk radius there is a qualitatively different behaviour. We test the theory by successfully estimating the positions of scattering resonances in geometries consisting of three and four small disks.

Final published version - some changes in the discussion and the labels on one figure are corrected

References in corpus (6)

• The Role of Diffraction in the Quantization of Dispersing Billiards
• Diffractive orbits in quantum billiards
• Edge Diffraction, Trace Formulae and the Cardioid Billiard
• Application of the diffraction trace formula to the three disk scattering system
• Semiclassical Quantisation Using Diffractive Orbits
• Geometric and Diffractive Orbits in the Scattering from Confocal Hyperbolae

Cited by in corpus (11)

• Scalar Casimir effect between Dirichlet spheres or a plate and a sphere
• Quantum corrections to the ground state energy of inhomogeneous neutron matter
• Unidirectional emission from circular dielectric microresonators with a point scatterer
• Casimir Interaction among Objects Immersed in a Fermionic Environment
• Quantization of Hyperbolic N-Sphere Scattering Systems in Three Dimensions
• Spectral properties of non-hydrogenic atoms in weak external fields
• Trace formula for a dielectric microdisk with a point scatterer
• Periodic orbit quantization of the Sinai billiard in the small scatterer limit
• Wave Chaos in Elastodynamic Cavity Scattering
• Negative length orbits in normal-superconductor billiard systems
• From chaotic to disordered systems - a periodic orbit approach