Related papers: Elimination and recursions in the scattering equat…
Our studies concern some aspects of scattering theory of the singular differential systems $ y'-x^{-1}Ay-q(x)y=\rho By, \ x>0 $ with $n\times n$ matrices $A,B, q(x), x\in(0,\infty)$, where $A,B$ are constant and $\rho$ is a spectral…
We introduce a new class of computationally tractable scattering problems in unbounded domains, which we call decomposable problems. In these decomposable problems, the computational domain can be split into a finite collection of…
The algebraic approach to the phase problem for the case of X-ray scattering from an ideal crystal is extended to the case of the neutron scattering, overcoming the difficulty related to the non-positivity of the scattering density. In this…
We lift the constraint of a diagonal representation of the Hamiltonian by searching for square integrable bases that support an infinite tridiagonal matrix representation of the wave operator. The class of solutions obtained as such…
For systems of polynomial equations, we study the problem of computing the Newton polytope of their eliminants. As was shown by Esterov and Khovanskii, such Newton polytopes are mixed fiber polytopes of the Newton polytopes of the input…
We consider the focusing nonlinear Schr\"odinger equation $i u_t + \Delta u + |u|^{p-1}u=0$, $p>1,$ and the generalized Hartree equation $iv_t + \Delta v + (|x|^{-(N-\gamma)}\ast |v|^p)|v|^{p-2}u=0$, $p\geq2$, $\gamma<N$, in the…
We consider a Schr{\"o}dinger equation with a nonlinearity which is a general perturbation of a power'' nonlinearity. We construct a profile decomposition adapted to this nonlinearity.We also prove global existence and scattering in a…
We show that the discrete operator stemming from the time and space discretization of evolutionary partial differential equations can be represented in terms of a single Sylvester matrix equation. A novel solution strategy that combines…
Extensions of standard one-dimensional supersymmetric quantum mechanics are discussed. Supercharges involving higher order derivatives are introduced leading to an algebra which incorporates a higher order polynomial in the Hamiltonian. We…
In this paper we firstly review how to \textit{explicitly} solve a system of $3$ \textit{first-order linear recursions }and outline the main properties of these solutions. Next, via a change of variables, we identify a class of systems of…
The purpose of this note is to give a mathematical explanation of a formula for the scattering matrix for a manifold with infinite cylindrical ends or a waveguide. This formula, which is well known in the physics literature, is sometimes…
In this paper, we prove a conjecture which was presented in a recent paper [Linear Algebra Appl. 2016; 496: 549--593]. We derive some practical necessary and sufficient conditions for the existence of a solution to a system of coupled…
A straightforward model for deposition and evaporation on discrete cells of a finite array of any dimension leads to a matrix equation involving a Sylvester-Kac type matrix. The eigenvalues and eigenvectors of the general matrix are…
In this paper, we consider a sorting scheme for potentially spurious scattering resonant pairs in one- and two-dimensional electromagnetic problems and in three-dimensional acoustic problems. The novel sorting scheme is based on a…
The different facets of the $R$-matrix method are presented pedagogically in a general framework. Two variants have been developed over the years: $(i)$ The "calculable" $R$-matrix method is a calculational tool to derive scattering…
A new computational method for solving the configuration-space Faddeev equations for the breakup scattering problem has been applied to nd scattering both below and above the two-body threshold.
We derive the (matrix-valued) Feynman rules of the mass perturbation theory and use it for the resummation of the $n$-point functions with the help of the Dyson-Schwinger equations. We use these results for a short review of the complete…
Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical…
Given polynomials a(z) of degree m and b(z) of degree n, we represent the inverse to the Sylvester resultant matrix of a(z) and b(z), if this inverse exists, as a canonical sum of m+n dyadic matrices each of which is a rational function of…
We study global behavior of small solutions of the Gross-Pitaevskii equation in three dimensions. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for large time, according to the linearized…