Exact, convergent periodic-orbit expansions of individual energy eigenvalues of regular quantum graphs

Abstract

We present exact, explicit, convergent periodic-orbit expansions for individual energy levels of regular quantum graphs. One simple application is the energy levels of a particle in a piecewise constant potential. Since the classical ray trajectories (including ray splitting) in such systems are strongly chaotic, this result provides the first explicit quantization of a classically chaotic system.

25 pages, 5 figures

References in corpus (5)

• Classical dynamics on graphs
• Exact trace formulae for a class of one-dimensional ray-splitting systems
• One-dimensional quantum chaos: Explicitly solvable cases
• Combinatorial identities for binary necklaces from exact ray-splitting trace formulae
• Mathematical Foundations of Regular Quantum Graphs

Cited by in corpus (12)

• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Green's function approach for quantum graphs: an overview
• The influence of geometry and topology of quantum graphs on their nonlinear-optical properties
• Explicit, analytical solution of scaling quantum graphs
• Unitary equivalence between the Green's function and Schrödinger approaches for quantum graphs
• Dressed quantum graphs with optical nonlinearities approaching the fundamental limit
• Solution of scaling quantum networks
• Weyl formulas for annular ray-splitting billiards
• Scaling and universality in nonlinear optical quantum graphs containing star motifs
• Quantum transport in randomized quantum graphs
• Scattering entropies of quantum graphs with several channels
• Quantum Systems with jump-discontinuous mass. I