Comment on "Recurrences without closed orbits"

In a recent paper Robicheaux and Shaw [Phys. Rev. A 58, 1043 (1998)] calculate the recurrence spectra of atoms in electric fields with non-vanishing angular momentum not equal to 0. Features are observed at scaled actions ``an order of magnitude shorter than for any classical closed orbit of this system.'' We investigate the transition from zero to nonzero angular momentum and demonstrate the existence of short closed orbits with L_z not equal to 0. The real and complex ``ghost'' orbits are created in bifurcations of the ``uphill'' and ``downhill'' orbit along the electric field axis, and can serve to interpret the observed features in the quantum recurrence spectra.Comment: 2 pages, 1 figure, REVTE

Semiclassical quantization with bifurcating orbits

Bifurcations of classical orbits introduce divergences into semiclassical spectra which have to be smoothed with the help of uniform approximations. We develop a technique to extract individual energy levels from semiclassical spectra involving uniform approximations. As a prototype example, the method is shown to yield excellent results for photo-absorption spectra for the hydrogen atom in an electric field in a spectral range where the abundance of bifurcations would render the standard closed-orbit formula without uniform approximations useless. Our method immediately applies to semiclassical trace formulae as well as closed-orbit theory and offers a general technique for the semiclassical quantization of arbitrary systems

Semiclassical Accuracy in Phase Space for Regular and Chaotic Dynamics

A phase-space semiclassical approximation valid to $O(\hbar)$ at short times is used to compare semiclassical accuracy for long-time and stationary observables in chaotic, stable, and mixed systems. Given the same level of semiclassical accuracy for the short time behavior, the squared semiclassical error in the chaotic system grows linearly in time, in contrast with quadratic growth in the classically stable system. In the chaotic system, the relative squared error at the Heisenberg time scales linearly with $\hbar_{\rm eff}$, allowing for unambiguous semiclassical determination of the eigenvalues and wave functions in the high-energy limit, while in the stable case the eigenvalue error always remains of the order of a mean level spacing. For a mixed classical phase space, eigenvalues associated with the chaotic sea can be semiclassically computed with greater accuracy than the ones associated with stable islands.Comment: 9 pages, 6 figures; to appear in Physical Review

Photoabsorption spectra of the diamagnetic hydrogen atom in the transition regime to chaos: Closed orbit theory with bifurcating orbits

With increasing energy the diamagnetic hydrogen atom undergoes a transition from regular to chaotic classical dynamics, and the closed orbits pass through various cascades of bifurcations. Closed orbit theory allows for the semiclassical calculation of photoabsorption spectra of the diamagnetic hydrogen atom. However, at the bifurcations the closed orbit contributions diverge. The singularities can be removed with the help of uniform semiclassical approximations which are constructed over a wide energy range for different types of codimension one and two catastrophes. Using the uniform approximations and applying the high-resolution harmonic inversion method we calculate fully resolved semiclassical photoabsorption spectra, i.e., individual eigenenergies and transition matrix elements at laboratory magnetic field strengths, and compare them with the results of exact quantum calculations.Comment: 26 pages, 9 figures, submitted to J. Phys.

Open circular billiards and the Riemann hypothesis

A comparison of escape rates from one and from two holes in an experimental container (e.g. a laser trap) can be used to obtain information about the dynamics inside the container. If this dynamics is simple enough one can hope to obtain exact formulas. Here we obtain exact formulas for escape from a circular billiard with one and with two holes. The corresponding quantities are expressed as sums over zeroes of the Riemann zeta function. Thus we demonstrate a direct connection between recent experiments and a major unsolved problem in mathematics, the Riemann hypothesis.Comment: 5 pages, 4 embedded postscript figures; v2: more explicit on how the Reimann Hypothesis arises from a comparison of one and two hole escape rate

Semiclassical quantization of the hydrogen atom in crossed electric and magnetic fields

The S-matrix theory formulation of closed-orbit theory recently proposed by Granger and Greene is extended to atoms in crossed electric and magnetic fields. We then present a semiclassical quantization of the hydrogen atom in crossed fields, which succeeds in resolving individual lines in the spectrum, but is restricted to the strongest lines of each n-manifold. By means of a detailed semiclassical analysis of the quantum spectrum, we demonstrate that it is the abundance of bifurcations of closed orbits that precludes the resolution of finer details. They necessitate the inclusion of uniform semiclassical approximations into the quantization process. Uniform approximations for the generic types of closed-orbit bifurcation are derived, and a general method for including them in a high-resolution semiclassical quantization is devised

Synthesis and alkyne-coupling chemistry of cyclomanganated 1- and 3-acetylindoles, 3-formylindole and analogues

The syntheses are reported of new cyclomanganated indole derivatives (1-acetyl-κO-indolyl-κC2)dicarbonylbis(trimethylphosphite)manganese (2), (1-methyl-3-acetyl-κO-indolyl-κC2)tetracarbonylmanganese (4), (3-formyl-κO-indolyl-κC2)tetracarbonylmanganese (5a) and (1-methyl-3-formyl-κO-indolyl-κC2)tetracarbonylmanganese (5b). The unusually complicated crystal structure of 5b has been determined, the first for a cyclomanganated aryl aldehyde. The preparations of a mitomycin-related pyrrolo-indole and related products by thermally promoted and oxidatively (Me3NO) initiated alkyne-coupling reactions of the previously known complex (1-acetyl-κO-indolyl-κC2)tetracarbonylmanganese (1) are reported for different alkynes and solvents. X-ray crystal structures are reported for the dimethyl acetylenedicarboxylate coupling product of 1 (dimethyl 1-methyl-l-hydroxypyrrolo[1,2a]-indole-2,3-dicarboxylate; 6a), and an unusually-cyclised triple insertion product 8 from the coupling of acetylene with 4, in which a cyclopentadiene moiety is η3-allyl-coordinated to Mn through only one double bond and an exocyclic carbon, but which rearranges on heating to an η5-cyclopentadienyl complex

Decimation and Harmonic Inversion of Periodic Orbit Signals

We present and compare three generically applicable signal processing methods for periodic orbit quantization via harmonic inversion of semiclassical recurrence functions. In a first step of each method, a band-limited decimated periodic orbit signal is obtained by analytical frequency windowing of the periodic orbit sum. In a second step, the frequencies and amplitudes of the decimated signal are determined by either Decimated Linear Predictor, Decimated Pade Approximant, or Decimated Signal Diagonalization. These techniques, which would have been numerically unstable without the windowing, provide numerically more accurate semiclassical spectra than does the filter-diagonalization method.Comment: 22 pages, 3 figures, submitted to J. Phys.

The hydrogen atom in an electric field: Closed-orbit theory with bifurcating orbits

Closed-orbit theory provides a general approach to the semiclassical description of photo-absorption spectra of arbitrary atoms in external fields, the simplest of which is the hydrogen atom in an electric field. Yet, despite its apparent simplicity, a semiclassical quantization of this system by means of closed-orbit theory has not been achieved so far. It is the aim of this paper to close that gap. We first present a detailed analytic study of the closed classical orbits and their bifurcations. We then derive a simple form of the uniform semiclassical approximation for the bifurcations that is suitable for an inclusion into a closed-orbit summation. By means of a generalized version of the semiclassical quantization by harmonic inversion, we succeed in calculating high-quality semiclassical spectra for the hydrogen atom in an electric field