Explicitly solvable model of Mössbauer scattering

An explicitly solvable model of Mössbauer scattering of rays by a nucleus bound in a harmonic-oscillator potential is constructed. The probability of elastic scattering, which is proportional to the Debye—Waller factor, is calculated in the framework of the explicitly solvable scattering problem. It is assumed that the rms deviation x of the nucleus and the photon wave numberk satisfykxE /E , whereE andE are typical energy levels of the photon and the oscillator states.St Petersburg State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 3, pp. 439–430, June, 1993.     

Space-time scattering for the Schrödinger equation

Results are obtained on the scattering theory for the Schrödinger equation $i\partial _t u(t,x) = - \Delta _x u(t,x) + V(t,x)u(t,x) + F(u(t,x))$ in spacesL ^r (R;L ^q (R ^d )) for a certain range ofr, q, the so-called space-time scattering. In the linear case (i.e.F≡)) the relation with usual configuration space scattering is established.     

Wave scattering by soft–hard three spaced waveguide

In this work we consider the sound radiation of a fundamental plane wave mode from a semi-infinite soft–hard duct. This duct is symmetrically located inside an infinite duct. This infinite waveguide consist of soft and hard plates. The whole system constitutes a three spaced waveguide. A closed form solution is obtained by using eigenfunction expansion matching method. This particular problem has been solved previously by Rawlins in closed form but without numerical work. Here the numerical results for reflection coefficient are given when the lowest mode propagates out from the semi-infinite duct. At the end we give comparison to both methods.     

The inside–outside duality for elastic scattering problems

In this article, we derive the inside–outside duality for two time-harmonic, elastic scattering problems. First, we consider a rigid scattering object inside an isotropic, homogeneous background medium and second, we consider a penetrable, inhomogeneous scattering object inside this background medium. For the first scattering problem, we make use of the particular behavior or certain eigenvalues of the corresponding far field operator to characterize interior Dirichlet eigenvalues of the negative Navier operator. Then we adapt this technique to determine interior transmission eigenvalues that correspond to the second scattering problem.     

Modified scattering for the critical nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation

$i{u}_t+\mathrm{\Delta }u=\lambda |u{|}^{\frac{2}{N}}u$

in all dimensions $N\ge 1$, where $\lambda \in \mathbb{C}$ and $?\lambda \le 0$. We construct a class of initial values for which the corresponding solution is global and decays as $t\to \infty$, like $t^{?\frac{N}{2}}$ if $?\lambda =0$ and like ${(t\log ?t)}^{?\frac{N}{2}}$ if $?\lambda <0$. Moreover, we give an asymptotic expansion of those solutions as $t\to \infty$. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at $u=0$. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents.     

Huygens’ principle and iterative methods in inverse obstacle scattering

The inverse problem we consider in this paper is to determine the shape of an obstacle from the knowledge of the far field pattern for scattering of time-harmonic plane waves. In the case of scattering from a sound-soft obstacle, we will interpret Huygens’ principle as a system of two integral equations, named data and field equation, for the unknown boundary of the scatterer and the induced surface flux, i.e., the unknown normal derivative of the total field on the boundary. Reflecting the ill-posedness of the inverse obstacle scattering problem these integral equations are ill-posed. They are linear with respect to the unknown flux and nonlinear with respect to the unknown boundary and offer, in principle, three immediate possibilities for their iterative solution via linearization and regularization. In addition to presenting new results on injectivity and dense range for the linearized operators, the main purpose of this paper is to establish and illuminate relations between these three solution methods based on Huygens’ principle in inverse obstacle scattering. Furthermore, we will exhibit connections and differences to the traditional regularized Newton type iterations as applied to the boundary to far field map, including alternatives for the implementation of these Newton iterations.     

Knauf’s degree and monodromy in planar potential scattering

We consider Hamiltonian systems on (T*?², dq ∧ dp) defined by a Hamiltonian function H of the “classical” form H = p ²/2 + V(q). A reasonable decay assumption V(q) → 0, ‖q‖ → ∞, allows one to compare a given distribution of initial conditions at t = ?∞ with their final distribution at t = +∞. To describe this Knauf introduced a topological invariant deg(E), which, for a nontrapping energy E > 0, is given by the degree of the scattering map. For rotationally symmetric potentials V(q) = W(‖q‖), scattering monodromy has been introduced independently as another topological invariant. In the present paper we demonstrate that, in the rotationally symmetric case, Knauf’s degree deg(E) and scattering monodromy are related to one another. Specifically, we show that scattering monodromy is given by the jump of the degree deg(E), which appears when the nontrapping energy E goes from low to high values.     

Foldy–Lax approximation on multiple scattering by many small scatterers

The multiple scattering of time harmonic wave emitted by a localized source through a medium with many scatterers can be approximated by an Foldy–Lax self-consistent system when the relative radius of each scatterer is small and the distribution of scatterers is sparse. The scattering amplitude in the Foldy–Lax self-consistent system will be specified in terms of scatterer volume and scattering strength. By neglecting the self-interaction effect, the difference from the exciting field in the Foldy–Lax formula to the analytic wave field given implicitly by the Lippmann–Schwinger integral equation is compared. An upper bound of the difference is obtained in terms of scaled radius and sparsity of the distribution of the scatterers.     

Inverse scattering problem with Levinson formula for eigenparameter-dependent discrete Sturm–Liouville equation

In this paper, an inverse scattering problem for discrete Sturm–Liouville equation with eigenparameter-dependent boundary condition is investigated. In quest of finding scattering function and the main equation of this problem, the uniqueness of the kernel is proven. Also, an appropriate Levinson-type formula based on the continuity of scattering function is given.     

Inverse problem of scattering theory for a class one-dimensional Schrödinger equation

Abstract

In this work, direct and inverse scattering problem on the real axis for the Schrödinger equation with piecewise-constant coefficient are studied. Using the new integral representations for solutions, the scattering data is defined, the main integral equations of the inverse scattering problem are obtained, the spectral characteristics of the scattering data are investigated and uniqueness theorem for the solution of inverse problem is proved.     

The scattering problem for the Schrödinger equation with a potential linear in time and in space. II. Correctness,smoothness, behavior of the solution at infinity

We study the scattering problem associated with the behavior of whispering gallery waves near the inflection point of the boundary. In order to solve the scattering problem, we prove the theorems of existence, uniqueness and smoothness of the solution. The formal asymptotic behavior is justified for t– and superexponential smallness of the wave field in the shadow zone is proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 13–29, 1985.     

Solving the non-isospectral Ablowitz–Ladik hierarchy via the inverse scattering transform and reductions

The non-isospectral Ablowitz–Ladik hierarchy is integrated by the inverse scattering transform. In contrast with the isospectral Ablowitz–Ladik hierarchy, the eigenvalues of the non-isospectral Ablowitz–Ladik equations in the scattering data are time-dependent. The multi-soliton solution for the hierarchy is presented. The reductions to the non-isospectral discrete NLS hierarchy and the non-isospectral discrete mKdV hierarchy and their solutions are considered.     

Convergence analysis of a high-order Nyström integral-equation method for surface scattering problems

In this paper we present a convergence analysis for the Nyström method proposed in [J Comput Phys 169 (1):80–110, 2001] for the solution of the combined boundary integral equation formulations of sound-soft acoustic scattering problems in three-dimensional space. This fast and efficient scheme combines FFT techniques and a polar change of variables that cancels out the kernel singularity. We establish the stability of the algorithms in the $L^2$ norm and we derive convergence estimates in both the $L^2$ and $L^\infty $ norms. In particular, our analysis establishes theoretically the previously observed super-algebraic convergence of the method in cases in which the right-hand side is smooth.     

Global well-posedness and scattering for a nonlinear Klein–Gordon system in low dimensions

This article investigates the global well-posedness and the scattering for a nonlinear Klein–Gordon system in spatial dimensions 1 and 2. We establish a Morawetz estimate for this system which is similar to the Morawetz estimate established by Nakanishi [K. Nakanishi, Energy scattering for nonlinear Klein–Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal. 169(1), pp. 201–225], combining this Morawetz estimate with the induction on energy argument developed by Bourgain [J. Bourgain, Global well-posedness of defocusing 3D critical NLS in the radial case, J. Am. Math. Soc. 12 (1999), pp. 145–171], the bound of a certain space-time norm and scattering result are obtained.