Analysis of oscillatory processes in lasers using the method of generalized eigenfunctions

The method of generalized eigenfunctions, which is used in the theory of diffraction, is applied to analyze stationary and narrow-band nonstationary processes in lasers. Using this method, one can avoid difficulties associated with integration of the eigenfunctions of an emitting system over the continuous spectrum, difficulties typical of the conventional frequency method. The method employs expansion in modes that are orthogonal inside the lasing medium. The problem of exponential growth of modes at infinity is eliminated. In addition, the field distribution inside the lasing medium is better described using the generalized eigenfunctions in a number of important cases.     

On the theory of eigenfunctions expansion of the Schrödinger operator

It is shown that the generalized eigenfunctions of the Schrödinger operator with singular potentials actins in L ₂(₃) are ordinary functions with determined asymptotic behaviour at infinity.     

Non-Relativistic Treatment of a Generalized Inverse Quadratic Yukawa Potential

A bound state solution is a quantum state solution of a particle subjected to a potential such that the particle's energy is less than the potential at both negative and positive infinity. The particle's energy may also be negative as the potential approaches zero at infinity. It is characterized by the discretized eigenvalues and eigenfunctions,which contain all the necessary information regarding the quantum systems under consideration. The bound state problems need to be extended using a more precise method and approximation scheme. This study focuses on the non-relativistic bound state solutions to the generalized inverse quadratic Yukawa potential. The expression for the non-relativistic energy eigenvalues and radial eigenfunctions are derived using proper quantization rule and formula method, respectively. The results reveal that both the ground and first excited energy eigenvalues depend largely on the angular momentum numbers, screening parameters, reduced mass, and the potential depth. The energy eigenvalues,angular momentum numbers,screening parameters,reduced mass,and the potential depth or potential coupling strength determine the nature of bound state of quantum particles. The explored model is also suitable for explaining both the bound and continuum states of quantum systems.     

Efficient computation of the spectrum of viscoelastic flows

The understanding of viscoelastic flows in many situations requires not only the steady state solution of the governing equations, but also its sensitivity to small perturbations. Linear stability analysis leads to a generalized eigenvalue problem (GEVP), whose numerical analysis may be challenging, even for Newtonian liquids, because the incompressibility constraint creates singularities that lead to non-physical eigenvalues at infinity. For viscoelastic flows, the difficulties increase due to the presence of continuous spectrum, related to the constitutive equations.The Couette flow of upper convected Maxwell (UCM) liquids has been used as a case study of the stability of viscoelastic flows. The spectrum consists of two discrete eigenvalues and a continuous segment with real part equal to ?1/We (We is the Weissenberg number). Most of the approximations in the literature were obtained using spectral expansions. The eigenvalues close to the continuous part of the spectrum show very slow convergence.In this work, the linear stability of Couette flow of a UCM liquid is studied using a finite element method. A new procedure to eliminate the eigenvalues at infinity from the GEVP is proposed. The procedure takes advantage of the structure of the matrices involved and avoids the computational overhead of the usual mapping techniques. The GEVP is transformed into a non-degenerate GEVP of dimension five times smaller. The computed eigenfunctions related to the continuous spectrum are in good agreement with the analytic solutions obtained by Graham [M.D. Graham, Effect of axial flow on viscoelastic Taylor–Couette instability, J. Fluid Mech. 360 (1998) 341].     

Filtering the eigenvalues at infinite from the linear stability analysis of incompressible flows

Steady state, two-dimensional flows may become unstable under two and three-dimensional disturbances if the flow parameters exceed some critical values. In many practical situations, determining the parameters at which the flow becomes unstable is essential. Linear hydrodynamic stability of a laminar flow leads to a generalized eigenvalue problem (GEVP) where the eigenvalues correspond to the rate of growth of the disturbances and the eigenfunctions to the amplitude of the perturbation. Solving GEVP’s is challenging, because the incompressibility of the liquid gives rise to singularities leading to non-physical eigenvalues at infinity that require substantial care. The high computational cost of solving the GEVP has probably discouraged the use of linear stability analysis of incompressible flows as a general engineering tool for design and optimization.In this work, we propose a new procedure to eliminate the eigenvalues at infinity from the GEVP associated to the linear stability analysis of incompressible flow. The procedure takes advantage of the structure of the matrices involved and avoids part of the computational effort of the standard mapping techniques used to compute the spectrum of incompressible flows. As an example, the method is applied in the solution of linear stability analysis of plane Couette flow.     

Rational Solutions of the Master Symmetries¶of the KdV Equation

In this article we characterize a certain class of rational solutions of the hierarchy of master symmetries for KdV. The result is that the generic rational potentials that decay at infinity and remain rational by all the flows of the master-symmetry KdV hierarchy are bispectral potentials for the Schr?dinger operator. By bispectral potentials we mean that the corresponding Schr?dinger operators possess families of eigenfunctions that are also eigenfunctions of a differential operator in the spectral variable. This complements certain results of Airault–McKean–Moser [4], Duistermaat–Grünbaum [10], and Magri–Zubelli [40]. As a consequence of bispectrality, the rational solutions of the master symmetries turn out to be solutions of a (generalized) string equation. Received: 28 January 1999 / Accepted: 22 October 1999     

Method of eigenfunctions of singular operators in the theory of diffraction by a thick vibrator

A rigorous electrodynamic solution of the problem of the diffraction of electromagnetic waves by the surface of a vibrator is described by a system of integrodifferential equations. The method of eigenfunctions of singular operators is used to reduce the basic system to an infinite algebraic Fredholm system of the second kind. The high efficiency of the proposed method is demonstrated on concrete examples. Zh. Tekh. Fiz. 68, 96–101 (April 1998)     

Dynamics of Gompertzian tumour growth under environmental fluctuations

We investigate the effect of correlated additive and multiplicative Gaussian white noise on the Gompertzian growth of tumours. Our results are obtained by solving numerically the time-dependent Fokker-Planck equation (FPE) associated with the stochastic dynamics. In our numerical approach we have adopted B-spline functions as a truncated basis to expand the approximated eigenfunctions. The eigenfunctions and eigenvalues obtained using this method are used to derive approximate solutions of the dynamics under study. We perform simulations to analyze various aspects, of the probability distribution, of the tumour cell populations in the transient- and steady-state regimes. More precisely, we are concerned mainly with the behaviour of the relaxation time (τ) to the steady-state distribution as a function of (i) of the correlation strength (λ) between the additive noise and the multiplicative noise and (ii) as a function of the multiplicative noise intensity (D) and additive noise intensity (α). It is observed that both the correlation strength and the intensities of additive and multiplicative noise, affect the relaxation time.     

Review: high pressure generation techniques beyond the limit of conventional diamond anvils

ABSTRACT

Current anvil designs and problems associated with various efforts to generate static high pressures beyond the limit of conventional diamond anvil cells (DACs) (～400?GPa) are reviewed. Pressures of up to 1?TPa have been reported by one research group using the double-stage DAC (ds-DAC) technique, but no other research group has successfully reproduced this high pressure result. Some research groups have used toroidal anvils, achieving pressures of >400?GPa. We have conducted numerous ds-DAC experiments and investigated the problems associated with such experiments. They include problems associated with various pressure scales in the multi-megabar region, difficulties in obtaining reliable X-ray diffraction patterns from micron-sized samples, and physical property measurements of tiny materials that may be harder than diamond. Each of these problems is discussed, following the summary of various experiments.     

Atom in electromagnetic field of near-atomic strength

Ionization and high-harmonics generation in a single hydrogen-like atom driven by a laser pulse of near-atomic field strength is the subject of this paper. We use exact solutions of the eigenvalue problem on particle motion in the cylindrically symmetric field (CSF) as a basis for the wave-function expansion. The superposition of the spherically symmetric intra-atomic field and linearly polarized laser field has the cylindrical symmetry. Hence, the use of the free-atom eigenfunctions as a basis for the wave-function expansion requires an infinitely increasing number of spherical harmonics (i.e., free-atom eigenfunctions) when the laser field strength approaches the intra-atomic field value. The eigenfunctions of the CSF problem depend on the laser field strength; therefore, the appropriate matrix elements show how the spectral width of atomic response and angular selection rules vary with increase in the laser field strength. The introduction gives a phenomenological semiclassical illustration of the problem. Talk presented at the oral issue of J. Russ. Laser Res. dedicated to the memory of Professor Vladimir A. Isakov, Professor Alexander S. Shumovsky, and Professor Andrei V. Vinogradov held in Moscow February 21–22, 2008.     

Quantum relativistic theory of an electron in terms of local observables

Dirac equation is reformulated in terms of real local observables, which are mean values of the wave function . The quadrivector current is shown to be a function of the potential vector and of other local observables. The equations describe the evolution of a four dimensional system T, X, Y, Z, and of two scalars, in the coordinate system ct, x, y, z. The current is proportional to the T vector. The Z vector is associated with the spin of the electron. Energy and gauge transformations correspond to rotations in the plane (X, Y). In the presence of a static field, the (real) solutions of the equations appear as eigenfunctions associated with energy eigenvalues. Received 7 September 1998     

Degeneracy of Resonances, Jordan Blocks, and Gamow-Jordan Eigenfunctions

We show that a degeneracy of resonances is associated with a second rank pole in the scattering matrix and a Jordan chain of generalized eigenfunctions of the radial Schrödinger equation. The generalized Gamow-Jordan eigenfunctions are basis elements of an expansion in complex resonance energy eigenfunctions. In this biorthonormal basis, any operator f(H _r which is a regular function of the Hamiltonian is represented by a nondiagonal complex matrix with a Jordan block of rank 2.     

Quantum motion over a finite one-dimensional domain: With and without dissipation

Quantum motion of a single particle over a finite one-dimensional spatial domain is considered for the generalized four parameter infinity of boundary conditions (GBC) of Carreauet al [1]. The boundary conditions permit complex eigenfunctions with nonzero current for discrete states. Explicit expressions are obtained for the eigenvalues and eigenfunctions. It is shown that these states go over to plane waves in the limit of the spatial domain becoming very large. Dissipation is introduced through Schrödinger-Langevin (SL) equation. The space and time parts of the SL equation are separated and the time part is solved exactly. The space part is converted to nonlinear ordinary differential equation. This is solved perturbatively consistent with the GBC. Various special cases are considered for illustrative purposes.     

The Riemann-Hilbert Formalism for Certain Linear and Nonlinear Integrable PDEs

Abstract

We show that by deforming the Riemann-Hilbert (RH) formalism associated with certain linear PDEs and using the so-called dressing method, it is possible to derive in an algorithmic way nonlinear integrable versions of these equations.

In the usual Dressing Method, one first postulates a matrix RH problem and then constructs dressing operators. Here we present an algorithmic construction of matrix Riemann-Hilbert (RH) problems appropriate for the dressing method as opposed to postulating them ad hoc. Furthermore, we introduce two mechanisms for the construction of the relevant dressing operators: The first uses operators with the same dispersive part, but with different decay at infinity, while the second uses pairs of operators corresponding to different Lax pairs of the same linear equation. As an application of our approach, we derive the NLS, derivative NLS, KdV, modified KdV and sine-Gordon equations.     

Scattering Matrices in Many-Body Scattering

In this paper we show that generalized eigenfunctions of many-body Hamiltonians H with short-range two-body interactions have distributional asymptotics at non-threshold channels. The leading terms of the asymptotics can be used to define a scattering matrix, and we show that this is the same (up to normalization) as that arising from the standard wave-operator approach. We also prove the existence of local distributional asymptotics for locally approximate generalized eigenfunctions in the more general setting of short range perturbations of a scattering metric, defined by Melrose in [13]. Received: 29 October 1997 / Accepted: 19 June 1998     

Optical vortices in the scattering field of magnetic domain holograms

Experimental investigations and theoretical-model analyses have been made of the magnetooptic diffraction of light at ferrite garnet magnetic films with a banded domain structure which includes isolated magnetic grating defects in the form of “forks” and “breaks.” An analysis of the magnetic grating structure and the light diffraction field shows that in terms of its action on laser radiation, a banded domain grating is similar to a computer-synthesized phase hologram of an isolated pure screw-wavefront dislocation. It is shown that as a result of magnetooptic diffraction at a magnetic hologram, optical vortices may be reconstructed with a helicoidal wavefront carrying the topological charge l=±1,±2. Zh. Tekh. Fiz. 68, 54–58 (December 1998)     

Resonant excitations of the 't Hooft-Polyakov monopole

The spherically symmetric magnetic monopole in an SU(2) gauge theory coupled to a massless Higgs field is shown to possess an infinite number of resonances or quasinormal modes. These modes are eigenfunctions of the isospin 1 perturbation equations with complex eigenvalues, E(n)=omega(n)-igamma(n), satisfying the outgoing radiation condition. For n--> infinity, their frequencies omega(n) approach the mass of the vector boson, M(W), while their lifetimes 1/gamma(n) tend to infinity. The response of the monopole to an arbitrary initial perturbation is largely determined by these resonant modes, whose collective effect leads to the formation of a long living breatherlike excitation with an amplitude decaying at late times as t(-5/6).     

Electric field controlled higher-order diffraction images of paraelectric potassium lithium tantalate niobate

We report some electric field controlled photorefractive higher-order diffraction phenomena of a paraelectric phase potassium lithium tantalate niobate crystal doped with iron. In experiments, a p-polarized semiconductor laser (532 nm) was used to record grating at a small incident angle. Higher-order diffraction images were observed when the signal beam was focused behind and in front of the crystal. Then the higher-order diffraction images were reconstructed by a p-polarized He–Ne laser (632.8 nm) in the direction perpendicular to the surface. The higher-order diffraction images could be controlled by the external electric field. A theory about the higher-order diffraction images of the K and 2K grating is developed. The results show that the even order diffraction images of the K grating and the odd order diffraction of the 2K grating overlap each other. The odd order diffraction images of the K grating are diffracted in unattached direction. The electric field controlled higher-order diffraction image provides a useful method for optical information processing.     

A new approach to the Schrödinger equation with rational potentials

A new analytic theory is established for the Schrödinger equation with a rational potential, including a complete classification of the regular eigenfunctions into three different types, an exact method of obtaining wavefunctions, an explicit formulation of the spectral equation (3 x 3 determinant) etc. All representations are exhibited in a unifying way via function-theoretic methods and therefore given in explicit form, in contrast to the prevailing discussion appealing to perturbation or variation methods or continued-fraction techniques. The irregular eigenfunctions at infinity can be obtained analogously and will be discussed separately as another solvable case for singular potentials.     

On the high-pressure phase transition in

X-ray diffraction (XRD) experiments have been carried out on quartz-like GaPO₄ at high pressure and room temperature. A transition to a high pressure disordered crystalline form occurs at 13.5 GPa. Slight heating using a YAG infrared laser was applied at 17 GPa in order to crystallize the phase in its stability field. The structure of this phase is orthorhombic with space group Cmcm. The cell parameters at the pressure of transition are a =7.306?, b =5.887? and c =5.124?. Received: 7 October 1997 / Received in final form: 17 November 1997 / Accepted: 18 November 1997