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We study the Lp-theory for the Schrödinger operator La with inverse-square potential a|x|-2. Our main result describes when Lp-based Sobolev spaces defined in terms of the operator (La)s/2 agree with those defined via (-Δ)s/2. We consider all regularities 0. In order to make the paper self-contained, we also review (with proofs) multiplier theorems, Littlewood-Paley theory, and Hardy-type inequalities associated to the operator La. [ABSTRACT FROM AUTHOR]

Using the Unified Transform, also known as the Fokas method, the solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann–Hilbert problem whose definition involves four spectral functions a , b , A , B . The functions a ( k ) and b ( k ) are defined via a nonlinear Fourier transform of the initial data, whereas A ( k ) and B ( k ) are defined via a nonlinear Fourier transform of the boundary values. In this paper, we provide an extensive study of these nonlinear Fourier transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values. The results can be used to determine the long-time asymptotics of the sine-Gordon quarter-plane solution via nonlinear steepest descent techniques. [ABSTRACT FROM AUTHOR]

We consider time discretizations of the Vlasov-HMF (Hamiltonian mean-field) equation based on splitting methods between the linear and nonlinear parts. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that the numerical solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping. Moreover, we prove that the modified state is close to the continuous one and provide error estimates with respect to the time step size. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded domain in R n , n ≥ 2 , and V ∈ L ∞ ( Ω ) be a potential function. Consider the following transmission eigenvalue problem for nontrivial v , w ∈ L 2 ( Ω ) and k ∈ R + , { ( Δ + k 2 ) v = 0 in Ω , ( Δ + k 2 ( 1 + V ) ) w = 0 in Ω , w − v ∈ H 0 2 ( Ω ) , ‖ v ‖ L 2 ( Ω ) = 1 . We show that the transmission eigenfunctions v and w carry the geometric information of supp ( V ) . Indeed, it is proved that v and w vanish near a corner point on ∂Ω in a generic situation where the corner possesses an interior angle less than π and the potential function V does not vanish at the corner point. This is the first quantitative result concerning the intrinsic property of transmission eigenfunctions and enriches the classical spectral theory for Dirichlet/Neumann Laplacian. We also discuss its implications to inverse scattering theory and invisibility. [ABSTRACT FROM AUTHOR]

We consider inverse obstacle and transmission scattering problems where the source of the incident waves is located on a smooth closed surface that is a boundary of a domain located outside of the obstacle/inhomogeneity of the media. The domain can be arbitrarily small but fixed.The scattered waves are measured on the same surface. An effective procedure is suggested for recovery of interior eigenvalues by these data. [ABSTRACT FROM AUTHOR]

In this paper a new numerical method for the shape reconstruction of obstacles in elastic scattering is proposed. Initially, the direct scattering problem for a rigid body and the mathematical setting for the corresponding inverse one are presented. Inverse uniqueness issues for the general case of mixed boundary conditions on the boundary of our obstacle, which are valid for a rigid body as well are established. The inversion algorithm based on the factorization method is presented into a suitable form and a new numerical scheme for the reconstruction of the shape of the scatterer, using far-field measurements, is given. In particular, an efficient Tikhonov parameter choice technique, calledImproved Maximum Product Criterion(IMPC) and its linchpin within the framework of the factorization method is exploited. Our regularization parameter is computed via a fast iterative algorithm which requires noa prioriknowledge of the noise level in the far-field data. Finally, the effectiveness of IMPC is illustrated with various numerical examples involving a kite, an acorn, and a peanut-shaped object. [ABSTRACT FROM PUBLISHER]

We study the inverse scattering problem with fixed observation angle data for a non-linear Schrödinger equation in 2D. We show that the main singularities of an unknown potential can be recovered from the Born approximation. Numerical examples are given to illustrate the method. [ABSTRACT FROM AUTHOR]

A general representation formula for the scattering matrix of a scattering system consisting of two self-adjoint operators in terms of an abstract operator valued Titchmarsh–Weyl m -function is proved. This result is applied to scattering problems for different self-adjoint realizations of Schrödinger operators on unbounded domains, Schrödinger operators with singular potentials supported on hypersurfaces, and orthogonal couplings of Schrödinger operators. In these applications the scattering matrix is expressed in an explicit form with the help of Dirichlet-to-Neumann maps. [ABSTRACT FROM AUTHOR]

We introduce and study Hardy spaces consisting of Clifford algebra-valued functions annihilated by perturbed Dirac operators in exterior uniformly rectifiable (UR) domains, and which radiate at infinity. In this context, we establish a higher dimensional version of Cauchy’s vanishing theorem, whose proof makes use of the properties of Cauchy-like operators in exterior UR domains, a sharp version of the Divergence Theorem in exterior Ahlfors regular domains, and a good understanding of the nature of various radiation conditions and properties of the far field pattern for Clifford algebra-valued null-solutions of the Helmholtz operator. [ABSTRACT FROM AUTHOR]

We study the nonlinear Schrödinger equation posed on product spaces , for and , with any k-dimensional compact Riemannian manifold. The main results concern global well-posedness and scattering for small data solutions in non-isotropic Sobolev fractional spaces. In the particular case of , -scattering is also obtained. [ABSTRACT FROM AUTHOR]

We study the nonlinear Schrödinger equation posed on product spaces , for and , with any k-dimensional compact Riemannian manifold. The main results concern global well-posedness and scattering for small data solutions in non-isotropic Sobolev fractional spaces. In the particular case of , -scattering is also obtained. [ABSTRACT FROM AUTHOR]

We consider the defocusing nonlinear Schrödinger equations i u +Δ u = | u| u with p being an even integer in dimensions d ≥ 5. We prove that an a priori bound of critical norm implies global well-posedness and scattering for the solution. [ABSTRACT FROM AUTHOR]

We consider inverse potential scattering problems where the source of the incident waves is located on a smooth closed surface outside of the inhomogeneity of the media. The scattered waves are measured on the same surface at a fixed value of the energy. We show that these data determine the bounded potential uniquely. [ABSTRACT FROM PUBLISHER]

We consider the radial defocusing nonlinear Schrödinger equations i u t + Δ u = | u | p u with supercritical exponent p > 4 in four space dimensions, and prove that any radial solution that remains bounded in the critical Sobolev space must be global and scatter. [ABSTRACT FROM AUTHOR]

The purpose of this work is to study the 3D focusing inhomogeneous nonlinear Schrödinger equation i u t + Δ u + | x | − b | u | 2 u = 0 , where 0 < b < 1 / 2 . Let Q be the ground state solution of − Q + Δ Q + | x | − b | Q | 2 Q = 0 and s c = ( 1 + b ) / 2 . We show that if the radial initial data u 0 belongs to H 1 ( R 3 ) and satisfies E ( u 0 ) s c M ( u 0 ) 1 − s c < E ( Q ) s c M ( Q ) 1 − s c and ‖ ∇ u 0 ‖ L 2 s c ‖ u 0 ‖ L 2 1 − s c < ‖ ∇ Q ‖ L 2 s c ‖ Q ‖ L 2 1 − s c , then the corresponding solution is global and scatters in H 1 ( R 3 ) . Our proof is based in the ideas introduced by Kenig–Merle [1] in their study of the energy-critical NLS and Holmer–Roudenko [2] for the radial 3D cubic NLS. [ABSTRACT FROM AUTHOR]

This paper is devoted to the discreteness of the transmission eigenvalue problems. It is known that this problem is not self-adjoint and a priori estimates are non-standard and do not hold in general. Two approaches are used. The first one is based on the multiplier technique and the second one is based on the Fourier analysis. The key point of the analysis is to establish the compactness and the uniqueness for Cauchy problems under various conditions. Using these approaches, we are able to rediscover quite a few known discreteness results in the literature and obtain various new results for which only the information near the boundary are required and there might be no contrast of the coefficients on the boundary. [ABSTRACT FROM AUTHOR]

Let L a be a Schrödinger operator with inverse square potential a | x | − 2 on R d , d ≥ 3 . The main aim of this paper is to prove weighted estimates for fractional powers of L a . The proof is based on weighted Hardy inequalities and weighted inequalities for square functions associated to L a . As an application, we obtain smoothing estimates regarding the propagator e i t L a . [ABSTRACT FROM AUTHOR]

We propose a new integral equation formulation to characterize and compute transmission eigenvalues for constant refractive index that play an important role in inverse scattering problems for penetrable media. As opposed to the recently developed approach by Cossonnière and Haddar [1,2] which relies on a two by two system of boundary integral equations our analysis is based on only one integral equation in terms of Dirichlet-to-Neumann or Robin-to-Dirichlet operators which results in a noticeable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further we employ the numerical algorithm for analytic non-linear eigenvalue problems that was recently proposed by Beyn [3] for the numerical computation of transmission eigenvalues via this new integral equation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the interior transmission eigenvalue problem for an inhomogeneous media with conductive boundary conditions. We prove the discreteness and existence of the transmission eigenvalues. We also investigate the inverse spectral problem of gaining information about the material properties from the transmission eigenvalues. In particular, we prove that the first transmission eigenvalue is a monotonic function of the refractive indexnand boundary conductivity parameter, and obtain a uniqueness result for constant coefficients. We provide some numerical examples to demonstrate the theoretical results in three dimensions. [ABSTRACT FROM AUTHOR]

We continue our investigation of transmission eigenvalues for spherically stratified media. In particular, in the case when the index of refractionn(r) is such that, we establish conditions onn(r) to guarantee that there exists an infinite number of complex transmission eigenvalues and when, we establish very weak conditions onn(r) to guarantee the existence of complex transmission eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper, we compare the far-field operators for the full nonlinear inverse scattering problem with the Born approximation as its linearization. The Factorization Method shows that both operators share the same behavior with respect to illposedness of the inverse problem. The results are derived for acoustic and electromagnetic scattering problems and the corresponding problem in electrical impedance tomography. [ABSTRACT FROM AUTHOR]

We introduce a class of hybrid boundary element methods for the solution of sound soft scattering problems in the exterior of two-dimensional smooth convex obstacles. To facilitate the applicability of our algorithms throughout the entire frequency spectrum, we have enriched our Galerkin approximation spaces, through incorporation of oscillations in the incident field of radiation, into the algebraic and trigonometric polynomial approximation spaces. The resulting methodologies have three distinctive properties. Indeed, from a theoretical point of view (1) they can be tuned to demand only an $$\mathcal {O}(k^{\varepsilon })$$ increase (for any $$\varepsilon >0)$$ in the number of degrees of freedom to maintain a fixed accuracy with increasing wavenumber k, owing to the optimal adaptation of approximation spaces to asymptotic stretching (shrinking) of illuminated and deep shadow regions (shadow boundaries), and (2) they are convergent for each fixed wavenumber k, thanks to the additional approximation spaces in the deep shadow region. Perhaps more importantly, from a practical point of view (3) they give rise to linear systems with significantly enhanced condition numbers and this, in turn, allows for more accurate solutions if desired. [ABSTRACT FROM AUTHOR]

We discuss a spectral asymptotics theory of an even zonal metric and a Schrödinger operator with zonal potentials on a sphere. We decompose the eigenvalue problem into a series of one-dimensional problems. We consider the individual behavior of this series of one-dimensional problems. We find certain Weyl's type of asymptotics on the eigenvalues. [ABSTRACT FROM AUTHOR]

Let H = − Δ + V be a Schrödinger operator on L 2 ( R n ) with real-valued potential V for n > 4 and let H 0 = − Δ . If V has sufficiently rapid decay, the wave operators defined as W ± = s − lim t → ± ∞ ⁡ e i t H e − i t H 0 are known to be bounded on L p ( R n ) for all 1 ≤ p ≤ ∞ if zero is not an eigenvalue, and on 1 < p < n 2 if zero is an eigenvalue. We show that these wave operators are also bounded on L 1 ( R n ) by direct examination of the integral kernel of the leading term. Furthermore, if ∫ R n V ( x ) ϕ ( x ) d x = 0 for all eigenfunctions ϕ , then the wave operators are L p bounded for 1 ≤ p < n . If, in addition ∫ R n x V ( x ) ϕ ( x ) d x = 0 , then the wave operators are bounded for 1 ≤ p < ∞ . [ABSTRACT FROM AUTHOR]

We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator $${\int\limits_{0} ^{\infty}{\rm e}^{iH_{0}t}\zeta(\frac{\vert x\vert }{R}) {\rm e}^{-iH_{0}t}{\rm d}t}$$ , as $${R \rightarrow \infty}$$ , is presented. Here, H is the free Dirac operator and $${\zeta\left(t\right)}$$ is such that $${\zeta\left(t\right) = 1}$$ for $${0 \leq t \leq 1}$$ and $${\zeta\left(t\right) = 0}$$ for $${t > 1}$$ . This approach allows us to obtain the time delay operator $${\delta \mathcal{T}\left(f\right)}$$ for initial states f in $${\mathcal{H} _{2}^{3/2+\varepsilon}(\mathbb{R}^{3};\mathbb{C}^{4})}$$ , $${\varepsilon > 0}$$ , the Sobolev space of order $${3/2+\varepsilon}$$ and weight 2. The relation between the time delay operator $${\delta\mathcal{T}\left(f\right)}$$ and the Eisenbud-Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented. [ABSTRACT FROM AUTHOR]

A numerical method to compute the scattering data for the Zakharov-Shabat system associated to the initial value problem for the nonlinear Schrödinger equation is proposed. The method involves the numerical solution of Volterra integral systems with structured kernels, of structured integral equations in unbounded domains and the identification of coefficients and parameters appearing in monomial-exponential sums. Numerical experiments confirm the effectiveness of the proposed technique. [ABSTRACT FROM AUTHOR]

We are interested in the localization of defects in non-homogeneous non-absorbing media, with far-field measurements, generated by plane waves. We present here a way to handle the inhomogeneous background problem by the means of a constrained optimization approach; in the lines of the inf criterion from the factorization method for constant backgrounds. We show that both cost function and constraints present a simple form that is easy to compute. They can thus be treated by many well-known optimization methods, and we numerically illustrate four of them. [ABSTRACT FROM PUBLISHER]

We construct a semiclassical parametrix for the resolvent of the Laplacian acting on functions on nontrapping conformally compact manifolds with variable sectional curvature at infinity. We apply this parametrix to analyze the Schwartz kernel of the semiclassical resolvent and Poisson operator and to show that the semiclassical scattering matrix is a semiclassical Fourier Integral Operator of appropriate class that quantizes the scattering relation. We also obtain high energy estimates for the resolvent and show existence of resonance free strips of arbitrary height away from the imaginary axis. We then use the results of Datchev and Vasy on gluing semiclassical resolvent estimates to obtain semiclassical resolvent estimates on certain conformally compact manifolds with hyperbolic trapping. [ABSTRACT FROM AUTHOR]

In this paper, the inverse scattering problem for the nonstationary Dirac system on the half-plane is considered. The uniqueness criterion for the inverse scattering problem in terms of boundary condition is described and the restoration of the potential from the scattering operator is proved. [ABSTRACT FROM AUTHOR]

This work studies the direct and inverse fixed energy scattering problem for the two-dimensional Schrödinger equation with a rather limited nonlinear index of refraction. In particular, using the Born approximation, we prove that all singularities of the unknown compactly supported potential belonging incan be obtained uniquely by the scattering data with fixed positive energy. The proof is based on the new estimates for the Faddeev Green’s function in. The main achievement here is the computation of the Born approximation, which is carried out using the total variation regularization method. Numerical examples with noisy data are given to illustrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

We consider the scattering theory for a pair of operatorsH0andH = H0 + VonL2(M,m), whereMis a Riemannian manifold,H0is a multiplication operator onM, andVis a pseudodifferential operator of order − μ, μ > 1. We show that a time-dependent scattering theory can be constructed, and the scattering matrix is a pseudodifferential operator on each energy surface. Moreover, the principal symbol of the scattering matrix is given by a Born approximation type function. The main motivation of the study comes from applications to discrete Schrödigner operators, and it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces. [ABSTRACT FROM AUTHOR]

Time-harmonic acoustic wave propagation in an inhomogeneous ocean with depth-dependent sound speed can be modeled by the Helmholtz equation in an infinite, three-dimensional (3D) waveguide of finite height. Using variational theory in Sobolev spaces we prove well posedness of the corresponding scattering problem from a bounded inhomogeneity inside such an ocean. To this end, we introduce an exterior Dirichlet-to-Neumann operator for depth-dependent sound speed and prove boundedness, coercivity, and holomorphic dependence of this operator in function spaces adapted to our weak solution theory. Analytic Fredholm theory then yields existence and uniqueness of solution for the scattering problem for all but a countable sequence of frequencies. The latter result generalizes corresponding theory for waveguide scattering with constant sound speed and easily extends to various related scattering problems, e.g. to scattering from impenetrable obstacles. [ABSTRACT FROM AUTHOR]

In this paper, we study the system of the coupled nonlinear Schröd-inger equation in dimension three. By establishing the a priori interaction, Morawetz estimate in the spirit of the work of Tao et al. [The nonlinear Schrödinger equation with combined power-type nonlinearities. Commun. Par. Differ. Equ. 2007;32:1281–1343], we prove the global well-posedness and scattering of the solutionto the defocusing coupled equation. Furthermore, we adapt the standard method from Cazenave [Semilinear Schrödinger equations. Vol. 10, Courant lecture notes in mathematics. Providence (RI): American Mathematical Society; 2003] to obtain the blowup result for the focusing case under the restriction: energy is negative and the initial data belongs toor the initial data is radial. [ABSTRACT FROM AUTHOR]

The research focuses on an inverse eigenvalue problem in an interior transmission phenomenon. There is an asymptotic structure on the eigenvalues. Given a sufficient quantity of transmission eigenvalues, we prove an uniqueness on the index of refraction in some inhomogeneous medium as an uniqueness problem in entire function theory. Interior transmission eigenvalues are intersection points of two entire functions. [ABSTRACT FROM AUTHOR]

We consider phaseless inverse scattering for the Schrödinger equation with compactly supported potential in dimension $$d\ge 2$$ . We give explicit formulas for solving this problem from appropriate data at high energies. As a corollary, we give also a global uniqueness result for this problem with appropriate data on a fixed energy neighborhood. [ABSTRACT FROM AUTHOR]

This work deals with the numerical computation of the inverse Born approximation associated with inverse scattering problems for the nonlinear Schrödinger equation in two space dimensions. We consider both backscattering and fixed angle data. The problem of computing the Born approximation is formulated as a linear inverse problem which is solved using a regularization method. Numerical examples with noisy data are given to illustrate the effectiveness of this method. [ABSTRACT FROM AUTHOR]