Yu. Dabaghian

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

We present exact, explicit, convergent periodic-orbit expansions for individual energy levels of regular quantum graphs in the paper. 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 an explicit quantization of a classically chaotic system.

Explicit analytical solution for scaling quantum graphs

We show that scaling quantum graphs with arbitrary topology are explicitly analytically solvable. This is surprising since quantum graphs are excellent models of quantum chaos and quantum chaotic systems are not usually explicitly analytically solvable.

Exact trace formulas for a class of one-dimensional ray-splitting systems

Using quantum graph theory we establish that the ray-splitting trace formula proposed by Couchman et al. [Phys. Rev. A 46, 6193 (1992)] is exact for a class of one-dimensional ray-splitting systems. Important applications in combinatorics are suggested.

One-Dimensional Quantum Chaos: Explicitly Solvable Cases

We present quantum graphs with remarkably regular spectral characteristics. We call them regular quantum graphs. Although regular quantum graphs are strongly chaotic in the classical limit, their quantum spectra are explicitly solvable in terms of periodic orbits. We present analytical solutions for the spectrum of regular quantum graphs in the form of explicit and exact periodic orbit expansions for each individual energy level.

Spectra of Regular Quantum Graphs

We consider a class of simple quasi-one-dimensional classically nonintegrable systems that capture the essence of the periodic orbit structure of general hyperbolic nonintegrable dynamical systems. Their behavior is sufficiently simple to allow a detailed investigation of both classical and quantum regimes. Despite their classical chaoticity, these systems exhibit a “nonintegrable analogue” of the Einstein-Brillouin-Keller quantization formula that provides their spectra explicitly, state by state, by means of convergent periodic orbit expansions.

Solution of scaling quantum networks

We show that all scaling quantum graphs are explicitly solvable, i.e., that any one of their spectral eigenvalues En is computable analytically, explicitly, and individually for any given n. This is surprising, since quantum graphs are excellent models of quantum chaos (see, e.g., T. Kottos and H. Schanz, Physica E 9, 523 (2001)).

Reply to Blümel's comment on 'Quantum chaos in elementary quantum mechanics'

We provide a short response to Blumels comment on our paper (Dabaghian and Jensen 2005 Eur. J. Phys. 26 423), concerning the periodic orbit summation techniques in the tunnelling regime.