Partial Recovery of a Potential from Backscattering Data

ABSTRACT

We prove that the main singularities,measured in the scale of Sobolev spaces,of the potential q in the Schrödinger Hamiltonian ?Δ+q,in dimensions n=2,3,are contained in the Born approximation for backscattering data.     

Space-time analytic smoothing effect for a system of nonlinear Schrödinger equations with non pseudo-conformally invariant interactions

In this study, we consider the local Cauchy problem for a system of nonlinear Schrödinger equations with non pseudo-conformally invariant interactions in the framework of space of charge and in the framework of space of energy. The main purpose of this study is to construct local solutions in function spaces of analytic vectors for the Galilei generator and the pseudo-conformal generator with data which satisfy exponentially decaying condition at spatial infinity. In particular, we improve the nonlinear estimates have been proved by Hayashi and Kato and Ozawa et al. involving the pseudo-conformal generator with coe?cient which depends on time of local existence of solutions and has singularity at finite value.     

Blow-up rate for critical nonlinear Schrödinger equation with Stark potential

We consider the blow-up solutions of the Cauchy problem for the critical nonlinear Schrödinger equation with Stark potential and the sharp lower and upper bounds of blow-up rate are established.     

Unique Continuation for Fractional Schrödinger Equations with Rough Potentials

This article deals with the weak and strong unique continuation principle for fractional Schrödinger equations with scaling-critical and rough potentials via Carleman estimates. Our methods extend to “variable coefficient” versions of fractional Schrödinger equations and operators on non-flat domains.     

Recovery of the singularities of a potential from backscattering data in general dimension

We prove that in dimension $n\ge 2$ the main singularities of a complex potential q having a certain a priori regularity are contained in the Born approximation $q_B$ constructed from backscattering data. This is archived using a new explicit formula for the multiple dispersion operators in the Fourier transform side. We also show that $q?q_B$ can be up to one derivative more regular than q in the Sobolev scale. On the other hand, we construct counterexamples showing that in general it is not possible to have more than one derivative gain, sometimes even strictly less, depending on the a priori regularity of q.     

Local Existence and WKB Approximation of Solutions to Schrödinger–Poisson System in the Two-Dimensional Whole Space

We consider the Schrödinger–Poisson system in the two-dimensional whole space. A new formula of solutions to the Poisson equation is used. Although the potential term solving the Poisson equation may grow at the spatial infinity, we show the unique existence of a time-local solution for data in the Sobolev spaces by an analysis of a quantum hydrodynamical system via a modified Madelung transform. This method has been used to justify the WKB approximation of solutions to several classes of nonlinear Schrödinger equation in the semiclassical limit.     

The Laguerre pseudospectral method for the radial Schrödinger equation

By transforming dependent and independent variables, radial Schrödinger equation is converted into a form resembling the Laguerre differential equation. Therefore, energy eigenvalues and wavefunctions of M-dimensional radial Schrödinger equation with a wide range of isotropic potentials are obtained numerically by using Laguerre pseudospectral methods. Comparison with the results from literature shows that the method is highly competitive.     

Multidimensional Schrödinger operators whose spectrum features a half-line and a Cantor set

We construct multidimensional Schrödinger operators with a spectrum that has no gaps at high energies and that is nowhere dense at low energies. This gives the first example for which this widely expected topological structure of the spectrum in the class of uniformly recurrent Schrödinger operators, namely the coexistence of a half-line and a Cantor-type structure, can be confirmed. Our construction uses Schrödinger operators with separable potentials that decompose into one-dimensional potentials generated by the Fibonacci sequence and relies on the study of such operators via the trace map and the Fricke-Vogt invariant. To show that the spectrum contains a half-line, we prove an abstract Bethe–Sommerfeld criterion for sums of Cantor sets which may be of independent interest.     

Uniqueness of the Modified Schrödinger Map in H 3/4+ε(ℝ2)

We establish the local well-posedness of the modified Schrödinger map in H ^3/4+ε(?²).     

Inverse Scattering for Schrödinger Operators with Miura Potentials,II. Different Riccati Representatives

This is the second in a series of papers on scattering theory for one-dimensional Schrödinger operators with Miura potentials admitting a Riccati representation of the form q = u′ + u ² for some u ∈ L ²(?). We consider potentials for which there exist ‘left’ and ‘right’ Riccati representatives with prescribed integrability on half-lines. This class includes all Faddeev–Marchenko potentials in L ¹(?, (1 + |x|)dx) generating positive Schrödinger operators as well as many distributional potentials with Dirac delta-functions and Coulomb-like singularities. We completely describe the corresponding set of reflection coefficients r and justify the algorithm reconstructing q from r.     

Local well-posedness of the nonlinear Schrödinger equations on the sphere for data in modulation spaces

In this paper, the authors discuss a priori estimates derived from the energy method to the initial value problem for the cubic nonlinear Schrödinger on the sphere S². Exploring suitable a priori estimates, the authors prove the existence of solution for data whose regularity is s = 1/4.     

Generic Controllability Properties for the Bilinear Schrödinger Equation

In [16 Chambrion , T. , Mason , P. , Sigalotti , M. , Boscain , U. ( 2009 ). Controllability of the discrete-spectrum Schrödinger equation driven by an external field . Ann. Inst. H. Poincaré Anal. Non Linéaire 26 : 329 – 349 . [Google Scholar]] we proposed a set of sufficient conditions for the approximate controllability of a discrete-spectrum bilinear Schrödinger equation. These conditions are expressed in terms of the controlled potential and of the eigenpairs of the uncontrolled Schrödinger operator. The aim of this paper is to show that these conditions are generic with respect to the uncontrolled and the controlled potential, denoted respectively by V and W. More precisely, we prove that the Schrödinger equation is approximately controllable generically with respect to W when V is fixed and also generically with respect to V when W is fixed and non-constant. The results are obtained by analytic perturbation arguments and through the study of asymptotic properties of eigenfunctions.