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Semiclassical Non-Trace Type Formulas for Matrix Element Fluctuations
and Weighted Densities of States
Semiclassical Non-Trace Type Formulas for Matrix Element Fluctuations
and Weighted Densities of States
1999-04-22
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arxiv.org/abs/chao-dyn/9904040v1
Phys. Rev. E 60 (1999) 1630
Nonlinear Sciences
Chaotic Dynamics
9 pages, 5 figures, REVTEX, submitted to Phys. Rev. E
Scientific paper
10.1103/PhysRevE.60.1630
Densities of states weighted with the diagonal matrix elements of two operators A and B, i.e., rho^(A,B)(E) = sum_n delta(E-E_n) cannot, in general, be written as a trace formula, and therefore no simple extension of semiclassical trace formulas is known for this case. However, from the high resolution analysis of quantum spectra in the semiclassical regime we find strong evidence that weighting the delta-functions in the quantum mechanical density of states with the product of diagonal matrix elements, , is equivalent to weighting the periodic orbit contributions in the semiclassical periodic orbit sum with the product of the periodic orbit means, _p_p, of the classical observables A and B. Results are presented for the hydrogen atom in a magnetic field for both the chaotic and near-integrable regime, and for the circle billiard.
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