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We consider generalizations of the Berry and Balazs nonspreading and accelerating solution of the time-dependent Schrodinger equation in empty space, which has been experimentally demonstrated in paraxial optics. In particular, we show that the original nonspreading wave packet is unstable. An explicit variation of the initial Airy-state evolves into the self-accelerating and self-compressing solution presented here. Quasi-diffraction-free finite energy Airy beams that are more realistic for experimental study are obtained by analytic continuation and their Wigner function is evaluated. Nonlinear generalizations related to second Painleve transcendents are briefly discussed.

In this Letter we consider n degrees-of-freedom integrable Hamiltonian systems subjected to a non-Hamiltonian perturbation controlled by a small parameter e . An obstruction to the analytic continuation of the integrals of motion of the unperturbed system with respect to e is developed for sufficiently small perturbations. The theory is applied to a perturbed system of Morse oscillators.

Two generalisations of the Evans function, for the analysis of the linearisation about solitary waves, are shown to be equivalent. The generalisation introduced by Alexander, Gardner and Jones [J. reine Angew. Math. 410 (1990) 167] is based on exterior algebra and the generalisation introduced by Swinton [Phys. Lett. A 163 (1992) 57] is based on a matrix formulation and adjoint systems. In regions of the complex plane where both formulations are defined, the equivalence is geometric: we show that the formulations are dual and the duality can be made explicit using Hodge duality and the Hodge star operator. Swinton's formulation excludes potential branch points at which the Alexander, Gardner and Jones formulation is well-defined. Therefore we consider the implications of equivalence on the analytic continuation of the two formulations.

The transition operator T for the scattering of a particle from N potentials V_j can be expanded into a series featuring the transition operators t_j associated with the individual potentials. For V_j(x) both absolutely and square integrable in x, we show, using an analytic continuation argument, that if T is on-shell, i.e. in < k|T(sigma^{2}+ i0)|k' >, |k|=|k'|=��, then each t_j is also on-shell., 7 pages, REVTeX

We explore the implications of the recently proposed Langevin quantization for quantum metastability, working within the semiclassical approximation. As far as we can see, the present treatment is simpler and more straightforward than the path integral approach. Indeed, no extra trick is needed and the correct result follows at once - as a consequence of general principles - from the representation of the propagator supplied by the Langevin quantization. Moreover, the imaginary part of the semiclassical propagator emerges naturally form the formalism and no analytic continuation has to be performed in order to make sense out of a divergent expression. Further applications of the strategy discussed in this Letter are pointed out.

The scattering problem on the Bruhat-Tits tree being the realization of the p-adic hyperbolic plane is studied. The spectral decomposition of the corresponding Laplace-Beltrami operator is constructed. The Lax-Phillips resonance scattering theory for the problem is developed in a closed form. The properties of the analytic continuation of the S-matrix are described.

Tunneling through a one-dimensional potential barrier is examined in the light of the supersymmetric WKB (SWKB) method. Comparison of the numerical values for the transmission coefficient ( T ) with the exact ones indicates that the SWKB method gives results much improved over those obtained by the usual WKB method. Analytic continuation of T for the inverted potential leads to the exact energy eigenvalues for a shape-invariant potential.

In this paper we consider the dependence of Schrodinger equation eigenvalue problems on the coupling-constant parameters in the potential. We show that unless great care is taken, analytic continuation in these parameters can lead to surprising and paradoxical conclusions. Careful analysis of the analytical properties shows that such eigenvalue problems can have elaborate internal structure; these problems can incorporate several different eigenvalue problems joined together. For example, the anharmonic oscillator, whose potential is V(x)=a2x6−3ax2, is actually four different eigenvalue problems combined into one.

When in the Birkhoff series the leading contributions of the main resonances (given by the continued fraction expansion of the linear frequency) are retained, the Fourier spectrum of the conjugation function is given by geometric series. Any Fourier coefficient corresponding to a main resonance is therefore approximated by a single pole in ϱ2, where ϱ is the distance from the origin in the normal coordinates, and the residue turns out to be exponentially small in 1/|ϱ| and proportional to the resonance width. In the ϱ2 plane the poles are on the real axis and their signs alternate since at any pole the winding number is resonant. The poles located on the real ϱ axis correspond to true resonances with island structure in the phase plane, the poles on the imaginary ϱ axis to virtual resonances with no counterpart in the phase plane. When the next to leading contribution of the main resonances are included, any Fourier coefficient is given by a series of poles with exponentially small residues. This singularity pattern is consistent with the asymptotic properties of the Birkhoff series and suggest that the Pade approximants should be used to continue them analytically. This is fully confirmed by numerical results.

The kinetic Ising model is analysed in the framework of the Migdal-Suzuki dynamical renormalization group, for arbitrary dimension. By analytic continuation, the case d =1+ ϵ is considered. Corrections to the d =1 results are calculated to leading order in ϵ.

Proofs are given that the quasidiscrete energy levels for the Stark effect can be obtained via quasiclassical minimization of inter-related spherically symmetrical hamiltonians. Implicit formulas, expressed just in terms of algebraic equations, are written down to first 1/ N order. The analytic continuation of such formulas towards resonance energies is also established. General symmetry properties, suitable parametrizations, and limiting cases are discussed.

Because of current interest in the chemical potential of N -electron systems, a definition is proposed for independent particles in one dimension. The method relies on analytic continuation of single-particle bound-state eigenvalues, infinite-simally close to integer N .

We show that the sturmian expansions of transition amplitudes for two-photon ionization may be handled as ordinary (single variable) power series, and we describe a simple method for performing their analytic continuation above the photoelectric threshold. Numerical results are presented for the hydrogenic states 1s, 2s, and 2p.

Solutions of the Henon-Heiles hamiltonian that are analytically continued into the complex time domain are found to possess a neutral boundary with a self-similar structure.

The averaged single particle Green function for electrons moving in a gaussian random potential is calculated by analytic continuation from the order parameter susceptibility of a thermodynamic system with order parameter dimension O. New results are obtained and discussed.

The positronium decay rate is calculated directly from the scattering amplitude by analytic continuation to the bound-state singularities. Relativistic wave function corrections are obtained which improve the agreement between theory and experiment.

It is shown that the use of hyperfunctions in classical field-theoretical calculations of radiation reaction terms leads to calculational simplifications. No analytic continuation is necessary. The calculation of electromagnetic radiation reaction terms is done as an example. The extension of hyperfunction techniques to general relativity and Yang-Mills fields is discussed.

A method of phase determination which involves an analytic continuation of the Fourier transform is described. The method has been successfully used to obtain the phases of some of the higher orders of nerve myelin.