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arXiv:quant-ph/0110109 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

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

R. Blümel, Yuri A. Dabaghian, Roderick V. Jensen

Published Apr 8, 2002• 26 citations
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Abstract

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.

DOI
10.1103/physreve.65.046222

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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.
We present an exact analytical solution of the spectral problem of quasi one-dimensional scaling quantum graphs. Strongly stochastic in the classical limit, these systems are frequently employed as models of quantum chaos. We show that despite their classical stochasticity all scaling quantum graphs are explicitly solvable in the form En=f(n)E_n=f(n)En​=f(n), where nnn is the sequence number of the energy level of the quantum graph and fff is a known function, which depends only on the physical and geometrical properties of the quantum graph. Our method of solution motivates a new classification scheme for quantum graphs: we show that each quantum graph can be uniquely assigned an integer mmm reflecting its level of complexity. We show that a taut string with piecewise constant mass density provides an experimentally realizable analogue system of scaling quantum graphs.
We present an exact analytical solution of the spectral problem of quasi-one-dimensional scaling quantum graphs. Strongly stochastic in the classical limit, these systems are frequently employed as models of quantum chaos. We show that despite their classical stochasticity all scaling quantum graphs are explicitly solvable in the form E(n) =f (n) , where n is the sequence number of the energy level of the quantum graph and f is a known function, which depends only on the physical and geometrical properties of the quantum graph. Our method of solution motivates a new classification scheme for quantum graphs: we show that each quantum graph can be uniquely assigned an integer m reflecting its level of complexity. We show that a network of taut strings with piecewise constant mass density provides an experimentally realizable analogue system of scaling quantum graphs.
Direct periodic-orbit expansions of individual spectral eigenvalues is a new direction in quantum mechanics. Using a unitary -matrix theory, we present exact, convergent, integral-free ghost orbit expansions of spectral eigenvalues for a step potential in the tunneling regime. We suggest an experiment to extract ghost orbit information from measured spectra in the tunneling regime (ghost orbit spectroscopy). We contrast our unitary, convergent theory with a recently published nonunitary, divergent theory [Yu. Dabaghian and R. Jensen, Eur. J. Phys. 26, 423 (2005)].
Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the periodic orbit sum and the uncertainty principle of the usual Fourier analysis, thus yielding results of high resolution and high precision. Based on the close analogy between periodic orbit trace formulae for regular and chaotic systems the technique is generalized in this paper for the semiclassical quantization of integrable systems. Thus, harmonic inversion is shown to be a universal tool which can be applied to a wide range of physical systems. The method is further generalized in two directions: firstly, the periodic orbit quantization will be extended to include higher order ~ corrections to the periodic orbit sum. Secondly, the use of cross-correlated periodic orbit sums allows us to significantly reduce the required number of orbi...