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First-Order ~ Corrections to Gutzwiller's Trace Formula for Systems with Discrete Symmetries

Holger Cartarius

Published Jan 1, 2006• 0 citations
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Abstract

In the reference (B. Gremaud, Phys. Rev. E 65, 056207 (2002)) flrst-order ~ corrections to Gutzwiller's trace formula for systems with a smooth potential were presented. Here we present an extension of that theory to systems with discrete symmetries. We apply the method to the two-dimensional hydrogen atom in a uniform magnetic fleld and exploit the C4v-symmetry in the calculation of the correction terms. The numerical results for the semiclassical values are compared with values extracted from exact quantum mechanical calculations. The comparison shows a very good agreement.

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In the reference (B. Gremaud, Phys. Rev. E 65, 056207 (2002)) flrst-order ~ corrections to Gutzwiller's trace formula for systems with a smooth potential were presented. Here we present an extension of that theory to systems with discrete symmetries. We apply the method to the two-dimensional hydrogen atom in a uniform magnetic fleld and exploit the C4v-symmetry in the calculation of the correction terms. The numerical results for the semiclassical values are compared with values extracted from exact quantum mechanical calculations. The comparison shows a very good agreement.
to Gutzwiller’s trace formula for systems with a smooth potential were presented. Here we present an extension of that theory to systems with discrete symmetries. We apply the method to the two-dimensional hydrogen atom in a uniform magnetic field and exploit the C4v-symmetry in the calculation of the correction terms. The numerical results for the semiclassical values are compared with values extracted from exact quantum mechanical calculations. The comparison shows a very good agreement. PACS numbers: 03.65.Sq, 05.45.Mt Keywords: semiclassical theories, Gutzwiller’s trace formula, � expansions, discrete symmetries, diamagnetic hydrogen atom 1.
While the scattering phase for several one-dimensional potentials can be exactly derived, less is known in multi-dimensional quantum systems. This work provides a method to extend the one-dimensional phase knowledge to multi-dimensional quantization rules. The extension is illustrated in the example of Bogomolny's transfer operator method applied in two quantum wells bounded by step potentials of different heights. This generalized semiclassical method accurately determines the energy spectrum of the systems, which indicates the substantial role of the proposed phase correction. Theoretically, the result can be extended to other semiclassical methods, such as Gutzwiller trace formula, dynamical zeta functions, and semiclassical Landauer–Büttiker formula. In practice, this recipe enhances the applicability of semiclassical methods to multi-dimensional quantum systems bounded by general soft potentials.
We derive contributions to the trace formula for the spectral density accounting for the role of diffractive orbits in two-dimensional billiard systems with corners. This is achieved by using the exact Sommerfeld solution for the Green function of a wedge. We obtain a uniformly valid formula for single-diffractive orbits which interpolates between formerly separate approaches (the geometrical theory of diffraction and Gutzwiller's trace formula). It yields excellent numerical agreement with exact quantum results, also in cases where other methods fail.