Use of Harmonic Inversion Techniques in the Periodic Orbit Quantization of Integrable Systems

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 hbar corrections to the periodic orbit sum. Secondly, the use of cross-correlated periodic orbit sums allows us to significantly reduce the required number of orbits for semiclassical quantization, i.e., to improve the efficiency of the semiclassical method. As a representative of regular systems, we choose the circle billiard, whose periodic orbits and quantum eigenvalues can easily be obtained.Comment: 21 pages, 9 figures, submitted to Eur. Phys. J.

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.

Manganese carbonyl-mediated reactions of azabutadienes with phenylacetylene, methyl acrylate and other unsaturated molecules

Reaction of PhCH₂Mn(CO)₅ with l,4-di-aryl-1-aza-1,3-butadienes gave substituted pyrrolinonyl rings which were η⁴-coordinated to a Mn(CO)₃ group. These are formed by intramolecular CO insertion into a (non-isolated) cyclomanganated intermediate, followed by cyclisation. Other unsaturated reagents (PhC≡CH, CH2=CHCOOMe, PhNCO) gave products arising from insertion of these, including a structurally characterised tri-aryl-η⁵-azacyclohexadienyl-Mn(CO)₃ complex from the reaction with the alkyne. PhCH₂Mn(CO)₅ reacts with l,4-di-aryl-1-aza-1,3-butadienes in the presence of unsaturated substrates to give products based on a cyclomanganated intermediate

2-(1,4-Dioxo-1,4-dihydro-2-naphthyl)-2-methylpropanoic acid

The sterically crowded title compound, C₁₄H₁₂O₄, crystallizes as centrosymmetric hydrogen-bonded dimers involving the carboxyl groups. The naphthoquinone ring system is folded by 11.5 (1)° about a vector joining the 1,4-C atoms, and the quinone O atoms are displaced from the ring plane, presumably because of steric interactions with the bulky substituent

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

Higher-order hbar corrections in the semiclassical quantization of chaotic billiards

In the periodic orbit quantization of physical systems, usually only the leading-order hbar contribution to the density of states is considered. Therefore, by construction, the eigenvalues following from semiclassical trace formulae generally agree with the exact quantum ones only to lowest order of hbar. In different theoretical work the trace formulae have been extended to higher orders of hbar. The problem remains, however, how to actually calculate eigenvalues from the extended trace formulae since, even with hbar corrections included, the periodic orbit sums still do not converge in the physical domain. For lowest-order semiclassical trace formulae the convergence problem can be elegantly, and universally, circumvented by application of the technique of harmonic inversion. In this paper we show how, for general scaling chaotic systems, also higher-order hbar corrections to the Gutzwiller formula can be included in the harmonic inversion scheme, and demonstrate that corrected semiclassical eigenvalues can be calculated despite the convergence problem. The method is applied to the open three-disk scattering system, as a prototype of a chaotic system.Comment: 14 pages, 6 figures, accepted for publication in Eur. Phys. J.