Exterior-Interior Duality for Discrete Graphs

Abstract

The Exterior-Interior duality expresses a deep connection between the Laplace spectrum in bounded and connected domains in ${\mathbb{R}}^2$, and the scattering matrices in the exterior of the domains. Here, this link is extended to the study of the spectrum of the discrete Laplacian on finite graphs. For this purpose, two methods are devised for associating scattering matrices to the graphs. The Exterior -Interior duality is derived for both methods.

15 pages 1 figure

References in corpus (6)

• Periodic Orbit Theory and Spectral Statistics for Quantum Graphs
• Quantum Graphs: A simple model for Chaotic Scattering
• Scattering on compact manifolds with infinitely thin horns
• Spectral Duality for Planar Billiards
• Quantization of Sinai's Billiard - A Scattering Approach
• Inside-Outside Duality for Planar Billiards -- A Numerical Study

Cited by in corpus (2)

• Even-odd flux quanta effect in the Fraunhofer oscillations of an edge-channel Josephson junction
• Nodal Statistics On Quantum Graphs