The inverse scattering problem at fixed energy for the three-dimensional Schrödinger equation with an exponentially decreasing potential

The relations between the Faddeev functions and the functions of classical scattering theory are found in the complex domain at fixed energy. For the three-dimensional case (without assumption of smallness of the potential) it is proved that the exponentially decreasing potential is uniquely determined by its scattering amplitude at fixed energy.     

A Schrödinger inverse scattering problem with a spectral dependence in the potential

The inverse scattering problem for the scalar Schrödinger equation , is considered. It is solved by reduction to the inverse scattering problem for a matrix Schrödinger equation: .This work has been done as part of the program Recherche Coopérative sur Programme No. 264: Etude interdisciplinaire des problèmes inverses.Physique Mathématique et Théorique, Equipe de recherche associée au CNRS, No. 154.     

Time-space fractional Schrödinger like equation with a nonlocal term

In this paper a time-space fractional Schr?dinger equation containing a nonlocal term has been studied. The time dependent solutions have been obtained in terms of the H-function. New general results include the results of integer Schr?dinger equation with a nonlocal term and the well-known quantum formulae for a free particle kernel.     

Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy

In this article we consider the Schrödinger operator inR ⁿ ,n3, with electric and magnetic potentials which decay exponentially as |x|. We show that the scattering amplitude at fixed positive energy determines the electric potential and the magnetic field.This research was supported by National Science Foundation Grant DMS93-05882.     

Pseudotime Schrödinger equation with absorbing potential for quantum scattering calculations

The Schr?dinger equation (Hpsi) (r) = [E+u(E)W(r)]psi(r) with an energy-dependent complex absorbing potential -u(E)W(r), associated with a scattering system, can be reduced for a special choice of u(E) to a harmonic inversion problem of a discrete pseudotime correlation function y(t) = phi(T)U(t)phi. An efficient formula for Green's function matrix elements is also derived. Since the exact propagation up to time 2t can be done with only approximately t real matrix-vector products, this gives an unprecedently efficient scheme for accurate calculations of quantum spectra for possibly very large systems.     

On equations solvable by the inverse scattering method for the Schrödinger operator

The new approach to deriving nonlinear evolution equations solvable by the inverse scattering method is proposed. In particular, this approach allows one to describe all equations solvable by the inverse scattering method for the Schrödinger operator.     

The inhomogeneous Schrödinger equation for the off-shell wave function

A method has been developed for calculating the off-shell wave function by solving the inhomogeneous Schrödinger equation with allowance for the nuclear and Coulomb interactions. The off-shell wave function makes it possible to construct the off-shell scattering amplitude in order to solve the problems for three or more particles. An important application of the method is the Trojan Horse calculations of nuclear reactions that are important in nuclear astrophysics. Specific calculations are performed for neutron and proton scattering on the ⁷Be nucleus. The Woods-Saxon potential is used and the spin-orbital interaction is taken into account.     

Spectral optimization for strongly singular Schrödinger operators with a star-shaped interaction

Letters in Mathematical Physics - We discuss the spectral properties of singular Schrödinger operators in three dimensions with the interaction supported by an equilateral star, finite or...     

Darboux transformations for the generalized Schrödinger equation

The intertwining operator technique is applied to the Schrödinger equation with an additional functional dependence h(r) on the right-hand side of the equation. The suggested generalized transformations turn into the Darboux transformations for both fixed and variable values of energy and angular momentum. A relation between the Darboux transformation and supersymmetry is considered.     

Quasilinear theory for the nonlinear Schrödinger equation with periodic coefficients

The nonlinear Schrödinger equation with periodic coefficients is analyzed under the condition of large variation in the local dispersion. The solution after n periods is represented as the sum of the solution to the linear part of the nonlinear Schrödinger equation and the nonlinear first-period correction multiplied by the number of periods n. An algorithm for calculating the quasilinear solution with arbitrary initial conditions is proposed. The nonlinear correction to the solution for a sequence of Gaussian pulses is obtained in the explicit form.     

The one-dimensional Schrödinger equation and a periodic potential

After recalling basic facts from the Titchmarsh-Weyl theory we derive and investigate the linear matrix equation, which holds for functions related to the spectral matrix of the one-dimensional periodic Schrödinger equation. The Weyl's solutions of the Schrödinger equation are used, when we solve this equation and associated nonlinear equations of the Milne's type. Two distinct trace formulae reconstructing the potential follow simply from the transformed and modified Milne's equations. Necessary spectral data of the inverse problem are determined by an infinite system of nonlinear first-order ordinary differential equations. Nonuniqueness of the solution of the inverse problem is confirmed on the other hand by writing a broad variety of the isospectral Darboux transformations.     

Solitons in a third-order nonlinear Schrödinger equation with the pseudo-Raman scattering and spatially decreasing second-order dispersion

Evolution of solitons is addressed in the framework of a third-order nonlinear Schrödinger equation (NLSE), including nonlinear dispersion, third-order dispersion and a pseudo-stimulated-Raman-scattering (pseudo- SRS) term, i.e., a spatial-domain counterpart of the SRS term, which is well known as a part of the temporal-domain NLSE in optics. In this context, it is induced by the underlying interaction of the high-frequency envelope wave with a damped low-frequency wave mode. In addition, spatial inhomogeneity of the second-order dispersion (SOD) is assumed. As a result, it is shown that the wavenumber downshift of solitons, caused by the pseudo-SRS, can be compensated with the upshift provided by decreasing SOD coefficients. Analytical results and numerical results are in a good agreement.     

New geometries associated with the nonlinear Schrödinger equation

We apply our recent formalism establishing new connections between the geometry of moving space curves and soliton equations, to the nonlinear Schr?dinger equation (NLS). We show that any given solution of the NLS gets associated with three distinct space curve evolutions. The tangent vector of the first of these curves, the binormal vector of the second and the normal vector of the third, are shown to satisfy the integrable Landau-Lifshitz (LL) equation = ×, ( = 1). These connections enable us to find the three surfaces swept out by the moving curves associated with the NLS. As an example, surfaces corresponding to a stationary envelope soliton solution of the NLS are obtained. Received 5 December 2001 Published online 2 October 2002 RID="a" ID="a"e-mail: radha@imsc.ernet.in     

Conservative method for simulation of a high-order nonlinear Schrdinger equation with a trapped term

We propose a new scheme for simulation of a high-order nonlinear Schrodinger equation with a trapped term by using the mid-point rule and Fourier pseudospectral method to approximate time and space derivatives, respectively. The method is proved to be both charge- and energy-conserved. Various numerical experiments for the equation in different cases are conducted. From the numerical evidence, we see the present method provides an accurate solution and conserves the discrete charge and energy invariants to machine accuracy which are consistent with the theoretical analysis.     

On the Schrödinger equation for the supersymmetric FRW model

We consider a time-dependent Schrödinger equation for the Friedmann–Robertson–Walker (FRW) model. We show that for this purpose it is possible to include an additional action invariant under reparametrization of time. The last one does not change the equations of motion for the minisuperspace model, but changes only the constraint. The same procedure is applied to the supersymmetric case.     

Exponentially small adiabatic invariant for the Schrödinger equation

We study an adiabatic invariant for the time-dependent Schrödinger equation which gives the transition probability across a gap from timet to timet. When the hamiltonian depends analytically on time, andt=–,t=+ we give sufficient conditions so that this adiabatic invariant tends to zero exponentially fast in the adiabatic limit.Supported by Fonds National Suisse de la Recherche, Grant 2000-5.600     

On the nonlinear Schrödinger equation for waves on a nonuniform current

A nonlinear Schrödinger equation with variable coefficients for surface waves on a large-scale steady nonuniform current has been derived without the assumption of a relative smallness of the velocity of the current. This equation can describe with good accuracy the loss of modulation stability of a wave coming to a counter current, leading to the formation of so-called rogue waves. Some theoretical estimates are compared to the numerical simulation with the exact equations for a two-dimensional potential motion of an ideal fluid with a free boundary over a nonuniform bottom at a nonzero average horizontal velocity.     

Global well-posedness for the Schrödinger equation coupled to a nonlinear oscillator

The Schrödinger equation with the nonlinearity concentrated at a single point proves to be an interesting and important model for the analysis of long-time behavior of solutions, including asymptotic stability of solitary waves and properties of weak global attractors. In this note, we prove global well-posedness of this system in the energy space H ¹.