Interaction of an atom with a laser field of intraatomic strength

The specific features of the nonlinear optical response of an atom at a laser pulse field of intraatomic strength have been investigated. An effective mathematical method, based on the expansion of the wave function in the basis of eigenfunctions of an axially symmetric boundary-value problem, is proposed to calculate the atomic response.     

Asymptotic method for determining the amplitude for three-particle breakup: Neutron-deuteron scattering

The process of neutron-deuteron scattering at energies above the deuteron-breakup threshold is described within the three-body formalism of Faddeev equations. Use is made of the method of solving Faddeev equations in configuration space on the basis of expanding wave-function components in the asymptotic region in bases of eigenfunctions of specially chosen operators. Asymptotically, wave-function components are represented in the form of an expansion in an orthonormalized basis of functions depending on the hyperangle. This basis makes it possible to orthogonalize the contributions of elastic-scattering and breakup channels. The proposed method permits determining scattering and breakup parameters from the asymptotic representation of the wave function without reconstructing it over the entire configuration space. The scattering and breakup amplitudes for states of total spin S = 1/2 and 3/2 were obtained for the s-wave Faddeev equation.     

Electrostatic solution and rayleigh approximation for small nonspherical particles in a spheroidal basis

The electrostatic problem for the case of axially symmetric particles is analyzed in a spheroidal basis. In this case, the wavenumber is zero and Maxwell’s equations are reduced to the Laplace equation for scalar potentials. An alternative approach involves solving integral equations that are similar to those obtained within the framework of the extended boundary conditions method. The scalar potentials are represented as expansions in terms of eigenfunctions of the Laplace equation in a spheroidal frame of reference, and unknown expansion coefficients are determined from an infinite set of linear algebraic equations (the separation of variables method). These two approaches yield exact solutions of the problem in the case of axially symmetric particles, which coincide with known solutions in particular cases. Investigation of infinite systems allowed finding the boundaries where these algorithms are valid. Numerical calculations showed that, for spheroidal Chebyshev particles (i.e., perturbed spheroids), the Rayleigh approximation based on the electrostatic solution is applicable in a wide range of the problem parameters and is in fair agreement with the results obtained using the discrete dipole approximation.     

Photoelectron angular distributions of molecules in bichromatic laser fields of circular polarization

Photoelectron angular distributions (PADs) from above-threshold ionization of O₂ and N₂ molecules irradiated by a bichromatic laser field of circular polarization are studied. The bichromatic laser field is specially modulated such that it can be used to mimic a sequence of one-cycle laser pulses. The PADs are greatly affected by the molecular alignment, the symmetry of the initial electronic distribution, and the carrier-envelope phase of the laser pulses. Generally, the PADs do not show any symmetry, and become symmetric about an axis only when the symmetric axis of laser field coincides with the symmetric axis of molecules. This study shows that the few-cycle laser pulses can be used to steer the photoelectrons and perform the selective ionization of molecules.     

Jack Polynomials in Superspace

This work initiates the study of orthogonal symmetric polynomials in superspace. Here we present two approaches leading to a family of orthogonal polynomials in superspace that generalize the Jack polynomials. The first approach relies on previous work by the authors in which eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland Hamiltonian were constructed. Orthogonal eigenfunctions are now obtained by diagonalizing the first nontrivial element of a bosonic tower of commuting conserved charges not containing this Hamiltonian. Quite remarkably, the expansion coefficients of these orthogonal eigenfunctions in the supermonomial basis are stable with respect to the number of variables. The second and more direct approach amounts to symmetrize products of non-symmetric Jack polynomials with monomials in the fermionic variables. This time, the orthogonality is inherited from the orthogonality of the non-symmetric Jack polynomials, and the value of the norm is given explicitly.     

Radiative properties of diamagnetic levels in atoms: Dependence of transition probability on magnetic field strength

We study the effect of diamagnetic interaction on the probability of radiative transitions of an atom from states split by a field. We write the analytic expressions for the diamagnetic corrections to the matrix elements of transitions belonging to the Lyman and Balmer series and of transitions between arbitrary nondegenerate states in hydrogen. We also discuss the perturbation theory for transitions from degenerate diamagnetic states. The theory is based on expanding in powers of the field strength the eigenfunctions and eigenvalues of the matrix of diamagnetic interaction in the subspace of states with given principal and magnetic quantum numbers. The field changes the coefficients in both the superposition and the degenerate basis. To derive the analytic expressions for the higher-order matrix elements, we use the Sturm expansion of the reduced Coulomb Green’s function. We also elaborate on the features of the frequency dependence of the corrections to the radiative matrix elements, which correlate with the structure of the diamagnetic spectrum of excited levels. Finally, we establish that the magnetic field acts selectively on the diamagnetic components of emission lines: as the field strength increases, an increase in the intensity of certain lines is accompanied by a decrease in the intensity of the other lines. Zh. éksp. Teor. Fiz. 116, 1161–1183 (October 1999)     

Use of a generalized method of eigenmodes in the theory of lasers

An analysis is made of the application a generalized method of eigenmodes, which is used in steady-state diffraction problems to the theory of lasers. In the study of open laser cavities the use of this method makes it possible to avoid difficulties associated with the exponential growth of the field at infinity. The generalized method developed previously for solving steady-state diffraction problems is extended to the case of non-steady-state narrow-band processes to construct in the ordinary way the equations describing the operation of the laser. The only difference is that the field is expanded in the eigenfunctions that are orthogonal in the region occupied by the active medium. This eliminates the difficulties associated with the behavior of the mode field at infinity. In addition, in a number of cases the generalized eigenfunctions (modes) correspond considerably better to the field distribution in the operating laser. Zh. Tekh. Fiz. 67, 46–49 (June 1997)     

Reissner--Nordström-de--Sitter-type Solution by a Gauge Theory of Gravity

We use the theory based on a gravitational gauge group （Wu＇s model） to obtain a spherical symmetric solution of the field equations for the gravitational potential on a Minkowski spacetime. The gauge group, the gauge covariant derivative, the strength tensor of the gauge feld, the gauge invariant Lagrangean with the cosmological constant, the field equations of the gauge potentiaIs with a gravitational energy-momentum tensor as well as with a tensor of the field of a point like source are determined. Finally, a Reissner-Nordstrom-de Sitter-type metric on the gauge group space is obtained.     

定跨度变截面梁的弯曲问题

一.引言 在处理梁的弯曲问题时,人们经常利用函数级数来表示有关的各量,并后而得到各该量的近似值。胡海昌曾经指出:在横向载荷和轴向力同时作用下,适宜于用梁的屈曲的本徵函数展开式来表示梁的挠度;其中φ_n是满足所给的梁的支座情况的屈曲本徵函数,a_n是常数系数。他求得一个相当简单的公式以已知的本徵函数和本徵值表示诸系数     

Complete and orthonormal eigenfunctions of excitations to a one-dimensional dark soliton

We study linear excitations to a one-dimensional dark soliton described by a defocusing nonlinear Schödinger equation. By solving an eigenvalue problem for the excitations we obtain all eigenvalues and eigenfunctions and prove rigorously that these eigenfunctions are orthonormal and form a complete set. We then use the eigenfunctions to obtain the exact form of linear excitations for any given initial condition and to investigate the transverse stability of the dark soliton. The rigorous results reported in the present work can be applied to study the dynamics of dark solitons in various nonlinear optical media and Bose-Einstein condensates.     

Coherent States for the Two-Dimensional Dirac-Moshinsky Oscillator Coupled to an External Magnetic Field

We show that the(2+l)-dimensional Dirac-Moshinsky oscillator coupled to an externa/ magnetic field can be treated algebraically with the SU(1,1) group theory and its group basis.We use the su(1,1) irreducible representation theory to find the energy spectrum and the eigenfunctions.Also,with the su(1,1) group basis we construct the relativistic coherent states in a closed form for this problem.     

On the quantization of solid body rotation

A general method for solving the problem of quantization of a top is proposed that allows for finding the Hamiltonian eigenfunctions in the form of polynomials of various degree n in the Cartesian coordinates with the use of the Lamé functions. All three coordinates x, y, and z are equivalently involved in computations, as well as their relations with ellipsoidal coordinates, which makes the computations symmetric.     

On a Class of Non-separable Quantum-Mechanical Eigenvalue Problems: Analytical and Technical Considerations within the Frame of a Born Expansion Method

An idea of Born is reviewed and elaborated to non-separable quantum-mechanical eigenvalue problems in which the Schrödinger equation can be solved exactly for a subconfiguration. (By subconfiguration we mean a subsystem in which one dynamic variable of the whole system is considered as parameter; derivations with respect to this variable are omitted.) The eigenfunctions in the subconfiguration (e.g., the eigenfunctions of a Born-Oppenheimer approximation) are used as a basis to expand the eigenfunction of the complete problem. By analytical methods it is shown how to construct the complete ensemble of solutions which can be systematically mapped and classified by their analytical behaviour in one of the singularities (in a regular singularity). A modification of the Numerov procedure is given to the numerical solution of the coupled second-order ordinary differential equations which arise from our treatment. The analytical asymptotic solutions are used to bridge over the asymptotic regions in which the error of the Numerov procedure is large. As a concrete example the comprehensive asymptotic analysis of the Schrödinger equation of a hydrogen-like ion in strong homogeneous magnetic field is presented, practical methods and computational aspects are discussed, and finally a few actual numerical results are reported: some energy levels are given as a function of field strength.     

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.     

A solution to the problem of the skin effect with a bias current in the Maxwell plasma by the method of expansion in eigenfunctions

A problem of the skin effect in the Maxwell plasma is solved analytically by the method of expansion in eigenfunctions based on the Vlasov–Maxwell kinetic equation with a self-consistent electric field. Specular electron reflection from the boundary is used as a boundary condition. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 33–38, February, 2009.     

Interbasis expansions for isotropic harmonic oscillator

The exact solutions of the isotropic harmonic oscillator are reviewed in Cartesian, cylindrical polar and spherical coordinates. The problem of interbasis expansions of the eigenfunctions is solved completely. The explicit expansion coefficients of the basis for given coordinates in terms of other two coordinates are presented for lower excited states. Such a property is occurred only for those degenerated states for given principal quantum number n.     

Reconciling the eigenmode analysis with the Maxwell-Bloch equations approach to superradiance in the linear regime

The superradiance from a slab of inverted two-level atoms is theoretically analyzed in the linear regime from both the perspective of the expansion in eigenfunctions of the integral equation with the Lienard-Wiechert potential as kernel, and that of linearizing the Maxwell-Bloch equations. We show the equivalence of both approaches. We show that the so-called Reduced Maxwell-Bloch equations do not yield even approximately the correct solution when applied in the obvious way, but that they can be made to give the correct solution by adding a fictitious input field.     

Surface plasmon modes in multi-layer thin-films

The surface plasmon modes in multi-layer thin-film structures of the metal-insulator-metal (MIM) type are modeled using a matrix method for the propagation of electromagnetic waves in stratified media. It is shown that the dispersion relation for the allowed surface plasmon modes is obtained from one of the matrix elements. The fundamental electromagnetic modes are the eigenfunctions of the differential equation for the magnetic field distribution and the eigenvalues are obtained from the dispersion relation. The expansion of an arbitrary wave profile in terms of the eigenfunctions is discussed and applied to the problem of surface plasmons propagating in a structure consisting of seven layers of alternating metal films and dielectrics.     

An initial-value method for integral operators—IV. Complex-valued kernels of laser theory

A new technique is presented for the calculation of the eigenvalues and eigenfunctions of complex-valued symmetric kernels which occur in laser theory. The method combines some classical results of integral equations and complex variables with a recent technique for transforming Fredholm integral equations into a Cauchy system of differential equations. The procedure is flexible and accurate, whereas many of the classical techniques, such as Rayleigh-Ritz, yield results of doubtful accuracy for the kernels of laser theory.