A dressing of zero-range potentials and electron-molecule scattering problem at Ramsauer-Townsend minimum

A dressing technique is used to improve zero range potential (ZRP) model. We consider a Darboux transformation starting with a ZRP, the result of the "dressing" gives a potential with non-zero range that depends on a seed solution parameters. Concepts of the partial waves and partial phases for non-spherical potential are used in order to perform Darboux transformation. The problem of scattering on the regular X$_{\hbox{n}}$ and YX$_{\hbox{n}}$ structures is studied. The results of the low-energy electron-molecule scattering on the dressed ZRPs are illustrated by model calculation for the configuration and parameters of the silane ($\hbox{SiH}_4$) molecule. \center{Key words: low-energy scattering, multiple scattering, Ramsauer-Townsend minimum, silane, zero range potential.Comment: 13 pages, 1 figur

Piecewise continuous partition function method in the theory of wave perturbations of inhomogeneous gas

The problem of wave disturbance propagation in rarefied gas in gravity field is explored. The system of hydrodynamic-type equations for a stratified gas in gravity field is derived from BGK equation by method of piecewise continuous partition function. The obtained system of the equations generalizes the Navier-Stokes at arbitrary density (Knudsen numbers). The verification of the model is made for a limiting case of a homogeneous medium. Results are in the good agreement with experiment and former theories at arbitrary Knudsen numbers.Comment: 12 pages, 5 figure

Piecewise continuous distribution function method: Fluid equations and wave disturbances at stratified gas

Wave disturbances of a stratified gas are studied. The description is built on a basis of the Bhatnagar -- Gross -- Krook (BGK) kinetic equation which is reduced down the level of fluid mechanics. The double momenta set is introduced inside a scheme of iterations of the equations operators, dividing the velocity space along and opposite gravity field direction. At both half-spaces the local equilibrium is supposed. As the result, the momenta system is derived. It reproduce Navier-Stokes and Barnett equations at the first and second order in high collision frequencies. The homogeneous background limit gives the known results obtained by direct kinetics applications by Loyalka and Cheng as the recent higher momentum fluid mechanics results of Chen, Rao and Spiegel. The ground state declines from exponential at the Knudsen regime. The WKB solutions for ultrasound in exponentially stratified medium are constructed in explicit form, evaluated and plotted.Comment: 20 pages, 7 figures, 14 ISNA conference, 199

General estimate of the first eigenvalue on manifolds

Ten sharp lower estimates of the first non-trivial eigenvalue of Laplacian on compact Riemannian manifolds are reviewed and compared. An improved variational formula, a general common estimate, and a new sharp one are added. The best lower estimates are now updated. The new estimates provide a global picture of what one can expect by our approach.Comment: 19 page

Von Neumann equations with time-dependent Hamiltonians and supersymmetric quantum mechanics

Starting with a time-independent Hamiltonian $h$ and an appropriately chosen solution of the von Neumann equation $i\dot\rho(t)=[ h,\rho(t)]$ we construct its binary-Darboux partner $h_1(t)$ and an exact scattering solution of $i\dot\rho_1(t)=[h_1(t),\rho_1(t)]$ where $h_1(t)$ is time-dependent and not isospectral to $h$. The method is analogous to supersymmetric quantum mechanics but is based on a different version of a Darboux transformation. We illustrate the technique by the example where $h$ corresponds to a 1-D harmonic oscillator. The resulting $h_1(t)$ represents a scattering of a soliton-like pulse on a three-level system.Comment: revtex, 3 eps file

Generalized Zero Range Potentials and Multi-Channel Electron-Molecule Scattering

A multi-channel scattering problem is studied from a point of view of integral equations system. The system appears while natural one-particle wave function equation of the electron under action of a potential with non-intersecting ranges is considered. Spherical functions basis expansion of the potentials introduces partial amplitudes and corresponding radial functions. The approach is generalized to multi-channel case by a matrix formulation in which a state vector component is associated with a scattering channel. The zero-range potentials naturally enter the scheme when the class of operators of multiplication is widen to distributions. %Analog of multipolar expansion is treated. Spin variables, o Oscillations and rotations are incorporated into the scheme.Comment: 11 pages, 1 figure, CEPAS2 con

Darboux Transformations for a Lax Integrable System in 2n2n-Dimensions

A $2n$-dimensional Lax integrable system is proposed by a set of specific spectral problems. It contains Takasaki equations, the self-dual Yang-Mills equations and its integrable hierarchy as examples. An explicit formulation of Darboux transformations is established for this Lax integrable system. The Vandermonde and generalized Cauchy determinant formulas lead to a description for deriving explicit solutions and thus some rational and analytic solutions are obtained.Comment: Latex, 14 pages, to be published in Lett. Math. Phy

Quantum feedback with weak measurements

The problem of feedback control of quantum systems by means of weak measurements is investigated in detail. When weak measurements are made on a set of identical quantum systems, the single-system density matrix can be determined to a high degree of accuracy while affecting each system only slightly. If this information is fed back into the systems by coherent operations, the single-system density matrix can be made to undergo an arbitrary nonlinear dynamics, including for example a dynamics governed by a nonlinear Schr\"odinger equation. We investigate the implications of such nonlinear quantum dynamics for various problems in quantum control and quantum information theory, including quantum computation. The nonlinear dynamics induced by weak quantum feedback could be used to create a novel form of quantum chaos in which the time evolution of the single-system wave function depends sensitively on initial conditions.Comment: 11 pages, TeX, replaced to incorporate suggestions of Asher Pere