103 research outputs found

### Detection of nano scale thin films with polarized neutron reflectometry at the presence of smooth and rough interfaces

By knowing the phase and modules of the reflection coefficient in neutron
reflectometry problems, a unique result for the scattering length density (SLD)
of a thin film can be determined which will lead to the exact determination of
type and thickness of the film. In the past decade, several methods have been
worked out to resolve the phase problem such as dwell time method, reference
layer method and variation of surroundings, among which the reference method
and variation of surroundings by using a magnetic substrate and polarized
neutrons is of the most applicability. All of these methods are based on the
solution of Schrodinger equation for a discontinuous and step-like potential at
each interface. As in real sample there are some smearing and roughness at
boundaries, consideration of smoothness and roughness of interfaces would
affect the final output result. In this paper, we have investigated the effects
of smoothness of interfaces on determination of the phase of reflection as well
as the retrieval process of the SLD, by using a smooth varying function (Eckart
potential). The effects of roughness of interfaces on the same parameters, have
also been investigated by random variation of the interface around it mean
position

### Explicit solutions to the Korteweg-de Vries equation on the half line

Certain explicit solutions to the Korteweg-de Vries equation in the first
quadrant of the $xt$-plane are presented. Such solutions involve algebraic
combinations of truly elementary functions, and their initial values correspond
to rational reflection coefficients in the associated Schr\"odinger equation.
In the reflectionless case such solutions reduce to pure $N$-soliton solutions.
An illustrative example is provided.Comment: 17 pages, no figure

### Complete determination of the reflection coefficient in neutron specular reflection by absorptive non-magnetic media

An experimental method is proposed which allows the complete determination of
the complex reflection coefficient for absorptive media for positive and
negative values of the momenta. It makes use of magnetic reference layers and
is a modification of a recently proposed technique for phase determination
based on polarization measurements. The complex reflection coefficient
resulting from a simulated application of the method is used for a
reconstruction of the scattering density profiles of absorptive non-magnetic
media by inversion.Comment: 14 pages, 4 figures, reformulation of abstract, ref.12 added,
typographical correction

### Time evolution of the scattering data for a fourth-order linear differential operator

The time evolution of the scattering and spectral data is obtained for the
differential operator $\displaystyle\frac{d^4}{dx^4} +\displaystyle\frac{d}{dx}
u(x,t)\displaystyle\frac{d}{dx}+v(x,t),$ where $u(x,t)$ and $v(x,t)$ are
real-valued potentials decaying exponentially as $x\to\pm\infty$ at each fixed
$t.$ The result is relevant in a crucial step of the inverse scattering
transform method that is used in solving the initial-value problem for a pair
of coupled nonlinear partial differential equations satisfied by $u(x,t)$ and
$v(x,t).$Comment: 19 page

### Inverse Spectral-Scattering Problem with Two Sets of Discrete Spectra for the Radial Schroedinger Equation

The Schroedinger equation on the half line is considered with a real-valued,
integrable potential having a finite first moment. It is shown that the
potential and the boundary conditions are uniquely determined by the data
containing the discrete eigenvalues for a boundary condition at the origin, the
continuous part of the spectral measure for that boundary condition, and a
subset of the discrete eigenvalues for a different boundary condition. This
result extends the celebrated two-spectrum uniqueness theorem of Borg and
Marchenko to the case where there is also a continuous spectru

### A unified approach to Darboux transformations

We analyze a certain class of integral equations related to Marchenko
equations and Gel'fand-Levitan equations associated with various systems of
ordinary differential operators. When the integral operator is perturbed by a
finite-rank perturbation, we explicitly evaluate the change in the solution. We
show how this result provides a unified approach to Darboux transformations
associated with various systems of ordinary differential operators. We
illustrate our theory by deriving the Darboux transformation for the
Zakharov-Shabat system and show how the potential and wave function change when
a discrete eigenvalue is added to the spectrum.Comment: final version that will appear in Inverse Problem

### The Response to a Perturbation in the Reflection Amplitude

We apply inverse scattering theory to calculate the functional derivative of
the potential $V(x)$ and wave function $\psi(x,k)$ of a one-dimensional
Schr\"odinger operator with respect to the reflection amplitude $r(k)$.Comment: 16 pages, no figure

### On the Two Spectra Inverse Problem for Semi-Infinite Jacobi Matrices

We present results on the unique reconstruction of a semi-infinite Jacobi
operator from the spectra of the operator with two different boundary
conditions. This is the discrete analogue of the Borg-Marchenko theorem for
Schr{\"o}dinger operators in the half-line. Furthermore, we give necessary and
sufficient conditions for two real sequences to be the spectra of a Jacobi
operator with different boundary conditions.Comment: In this slightly revised version we have reworded some of the
theorems, and we updated two reference

### Exact solutions to the focusing nonlinear Schrodinger equation

A method is given to construct globally analytic (in space and time) exact
solutions to the focusing cubic nonlinear Schrodinger equation on the line. An
explicit formula and its equivalents are presented to express such exact
solutions in a compact form in terms of matrix exponentials. Such exact
solutions can alternatively be written explicitly as algebraic combinations of
exponential, trigonometric, and polynomial functions of the spatial and
temporal coordinates.Comment: 60 pages, 18 figure

### The Two-Spectra Inverse Problem for Semi-Infinite Jacobi Matrices in The Limit-Circle Case

We present a technique for reconstructing a semi-infinite Jacobi operator in
the limit circle case from the spectra of two different self-adjoint
extensions. Moreover, we give necessary and sufficient conditions for two real
sequences to be the spectra of two different self-adjoint extensions of a
Jacobi operator in the limit circle case.Comment: 26 pages. Changes in the presentation of some result

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