Semiclassical, trajectory-coherent, quasi-energy states for the one-dimensional schrödinger equation

On the basis of the complex WKB-Maslov method, semiclassical, trajectory-coherent, quasi-energy states are constructed for the one-dimensional Schrödinger equation with the potential corresponding to the well-known classical Duffing equation with a time-periodic driving force. Classical objects required to construct semiclassical quasi-energy states are determined by converging series in powers of a small parameter included in the potential. Leading terms of these series are calculated.     

The semiclassical functional equation

A semiclassical analog of the functional equation for the Riemann zeta function is considered. In the case of the zeta function itself, this equation forms the basis for a finite approximation to the Dirichlet series, known as the approximate functional equation. In the same way, the semiclassical functional equation can be shown to give rise to a finite approximation to the semiclassical representation of the quantum spectral determinant as a sum over classical pseudo-orbits. This finite approximation has been called the Riemann-Siegel look-alike formula. The formal nature of the derivation of this result is discussed and the fact that it appears to imply a remarkable relationship between long and short pseudo-orbits is shown.     

Periodic-orbit theory of level correlations

We present a semiclassical explanation of the so-called Bohigas-Giannoni-Schmit conjecture which asserts universality of spectral fluctuations in chaotic dynamics. We work with a generating function whose semiclassical limit is determined by quadruplets of sets of periodic orbits. The asymptotic expansions of both the nonoscillatory and the oscillatory part of the universal spectral correlator are obtained. Borel summation of the series reproduces the exact correlator of random-matrix theory.     

Semiclassical Eigenfunctions of the Schrödinger Operator on a Graph That Are Localized Near a Subgraph

We study semiclassical eigenvalues and eigenfunctions of the Schrödinger operator on a geometric graph. We show that nontrivial boundary conditions at vertices lead to the existence of eigenfunctions, concentrated near a single vertex. We also construct semiclassical eigenfunctions, localized near edges and discuss general construction of spectral series which correspond to a general subgraph.     

A unified semiclassical description of elastic scattering by a central field

The diffraction-integral formulation of the semiclassical limit of the quantal wavefunction, as proposed in an earlier paper, is applied to the treatment of elastic scattering by a general central-field potential. Angular-momentum quantisation is shown to be a natural consequence of the theory and partial-wave series representations of the s-matrix and asymptotic wavefunction are derived. In addition, the method and results establish a connection between refractive, diffractive, dynamical and uniform-approximation theories of semiclassical scattering.     

Theory of forward glory scattering for chemical reactions: New derivation of a uniform semiclassical formula for the scattering amplitude

The theory of forward glory scattering is investigated for a state-to-state chemical reaction whose scattering amplitude can be written as a Legendre partial wave series. Legendre series occur in the exact quantum theory of reactive scattering when the initial and final helicity quantum numbers are zero, as well as in many approximate theories of chemical reactions. The starting point for the semiclassical theory is a two-dimensional integral representation for the scattering amplitude. A uniform semiclassical approximation is derived that is valid for angles both on, and off, the axial caustic associated with the glory. The derivation is the first application to a concrete problem in molecular physics of a method outlined by J. N. L. Connor and H. R. Mayne in 1979 for the uniform semiclassical evaluation of multidimensional integrals. The approach exploits the theory of singularities of differential mappings. The key step in the derivation is an exact one-to-one change of variables in the neighbourhood of the stationary phase points that locally reduce the two-dimensional phase of the integrand to a non-polynomial canonical form. The derivation complements a different semiclassical glory analysis reported in a companion paper.     

Semiclassical theory for many-body fermionic systems

We present a treatment of many-body fermionic systems that facilitates an expression of well-known quantities in a series expansion inħ. The ensuing semiclassical result contains, to a leading order of the response function, the classical time correlation function of the observable followed by the Weyl-Wigner series; on top of these terms are the periodic-orbit correction terms. The treatment given here starts from linear response assumption of the many-body theory and in its connection with semiclassical theory, it assumes that the one-body quantal system has a classically chaotic dynamics. Applications of the framework are also discussed.     

Bifurcation phenomena of photodetached electron flux in parallel external fields

A semiclassical method based on the closed-orbit theory is applied to analysing the dynamics of photodetached electron of H$^- $ in the parallel electric and magnetic fields. By simply varying the magnetic field we reveal spatial bifurcations of electron orbits at a fixed emission energy, which is referred to as the fold caustic in classical motion. The quantum manifestations of these singularities display a series of intermittent divergences in electronic flux distributions. We introduce semiclassical uniform approximation to repair the electron wavefunctions locally in a mixed phase space and obtain reasonable results. The approximation provides a better treatment of the problem.     

Semiclassical ordering in the large-N pyrochlore antiferromagnet

We study the semiclassical limit of the Sp(N) generalization of the pyrochlore lattice Heisenberg antiferromagnet by expanding about the N --> infinity saddlepoint in powers of a generalized inverse spin. To leading order, we write down an effective Hamiltonian as a series in loops on the lattice. Using this as a formula for calculating the energy of any classical ground state, we perform Monte Carlo simulations and find a unique collinear ground state. This state is not a ground state of linear spin-wave theory, and can therefore not be a physical (N = 1) semiclassical ground state.     

Stark broadening of neutral potassium lines

Using a semiclassical approach, we have calculated electron-proton and Ar II impact line widths and shifts of 50 neutral potassium lines. The comprehensive set of results obtained is used for investigation of Stark-broadening-parameter regularities within the spectral series.     

Spectral series of the Schrödinger operator with delta-potential on a three-dimensional spherically symmetric manifold

The spectral series of the Schrödinger operator with a delta-potential on a threedimensional compact spherically symmetric manifold in the semiclassical limit as h → 0 are described.     

Semiclassical propagator and van Vleck determinant in a mixed position--momentum space

In this paper a semiclassical propagator in a mixed position--momentum space is derived in the formalism of Maslov's multi-dimensional semiclassical theory. The corresponding mixed van Vleck determinant is also given explicitly. The propagator can be used to locally fix semiclassical divergences in singular regions of configuration space. It is shown that when a semiclassical propagator is transformed from one representation to another, its form is invariant.     

Temperature dependent pressure induced lineshape of O3 rotational transitions in air

The pressure induced broadening of a series of pure rotational transitions of ozone have been measured as a function of temperature. Results of experiments are compared with calculations employing the complex semiclassical theory of Robert and Bonamy. This set of rotational transitions is the dominant feature of the millimeter and submillimeter ozone spectra to be measured in the upcoming EOS-MLS mission.     

Towards Nonlinear Quantum Fokker-Planck Equations

It is demonstrated how the equilibrium semiclassical approach of Coffey et al. can be improved to describe more correctly the evolution. As a result a new semiclassical Klein-Kramers equation for the Wigner function is derived, which remains quantum for a free quantum Brownian particle as well. It is transformed to a semiclassical Smoluchowski equation, which leads to our semiclassical generalization of the classical Einstein law of Brownian motion derived before. A possibility is discussed how to extend these semiclassical equations to nonlinear quantum Fokker-Planck equations based on the Fisher information.     

Stark broadening of neutral sodium lines

A semiclassical approach has been used to evaluate Stark broadening of atomic lines and also electron-and proton-impact line widths and shifts of 30 neutral sodium lines. The results are used to investigate Stark broadening-parameter regularities within the spectral series.     

Stark broadening of neutral helium lines

Using a semiclassical approach for the Stark broadening of atomic lines, we have calculated electron and proton impact line widths and shifts of 56 neutral He lines in the ultraviolet, visible and i.r. region of the spectrum. The comprehensive set of results obtained is used for investigation of Stark-broadening parameter regularities within the spectral series.     

Multi-instanton contributions in quantum mechanics

We calculate multi-instanton contributions in quantum mechanics, at leading order in the semiclassical limit. Our results are consistent with the numerical information available for the double-well and periodic cosinus potential. We briefly discuss the relevance of our results to the problem of summation of the perturbation series.