Strichartz Estimates for the Schrödinger Equation on Polygonal Domains

We prove Strichartz estimates with a loss of derivatives for the Schrödinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates on the polygon follow from those on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz estimates then follow from a recent result of the second author regarding the Schrödinger equation on the Euclidean cone.     

Eigenfunction Expansions for Schrödinger Operators on Metric Graphs

We construct an expansion in generalized eigenfunctions for Schr?dinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.      

Global Superconvergence Estimates of Finite Element Method for Schrödinger Equation

In this paper, we shall study the initial boundary value problem of Schrödinger equation. The second order gradient superconvergence estimates for the problem are obtained solving by linear finite elements.     

Strichartz Estimates for Schrödinger Equations on Scattering Manifolds

The present article is concerned with Schrödinger equations on non-compact Riemannian manifolds with asymptotically conic ends. It is shown that, for any admissible pair (including the endpoint), local in time Strichartz estimates outside a large compact set are centered at origin hold. Moreover, we prove global in space Strichartz estimates under the nontrapping condition on the metric.     

Dispersive Estimates of Solutions to the Schrödinger Equation

We prove time decay L¹ → L^∞ estimates for the Schr?dinger group e^{it(−Δ + V)} for real-valued potentials satisfying V (x) = O (|x|^−δ), |x| ≫ 1, with δ > 5/2. Communicated by Bernard Helffer submitted 27/11/04, accepted 29/04/05     

Dispersion Estimates for Spherical Schrödinger Equations

We derive a dispersion estimate for one-dimensional perturbed radial Schrödinger operators. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum.     

Strichartz Estimates for Non-Elliptic Schrödinger Equations on Compact Manifolds

In this note we consider the Schrödinger equation on compact manifolds equipped with possibly degenerate metrics. We prove Strichartz estimates with a loss of derivatives. The rate of loss of derivatives depends on the degeneracy of metrics. For the non-degenerate case we obtain, as an application of the main result, the same Strichartz estimates as that in the elliptic case. This extends Strichartz estimates for Riemannian metrics proved by Burq-Gérard-Tzvetkov to the non-elliptic case and improves the result by Salort for the degenerate case. We also investigate the optimality of the result for the case on 𝕊³ × 𝕊³.     

Sharp Estimates for Schrödinger Groups on Hardy Spaces for $$0

Journal of Fourier Analysis and Applications - In this paper, we discuss optimal constants and extremisers of Kato-smoothing estimates for the 2D Dirac equation. Smoothing estimates are...     

Inverse Spectral Problem for the Schrödinger Equation with an Additional Linear Potential

Theoretical and Mathematical Physics - We consider the one-dimensional Schrödinger equation with an additional linear potential on the whole axis and construct a transformation operator with a...     

The Nonlinear Schrödinger Equation on Hyperbolic Space

In this article we study some aspects of dispersive and concentration phenomena for the Schrödinger equation posed on hyperbolic space  ⁿ , in order to see if the negative curvature of the manifold gets the dynamics more stable than in the Euclidean case. It is indeed the case for the dispersive properties: we prove that the dispersion inequality is valid, in a stronger form than the one on ? ⁿ . However, the geometry does not have enough of an effect to avoid the concentration phenomena and the picture is actually worse than expected. The critical nonlinearity power for blow-up turns out to be the same as in the euclidean case, and we prove that there are more explosive solutions for critical and supercritical nonlinearities.     

Effective Approximation for the Semiclassical Schrödinger Equation

The computation of the semiclassical Schrödinger equation presents major challenges because of the presence of a small parameter. Assuming periodic boundary conditions, the standard approach consists of semi-discretisation with a spectral method, followed by an exponential splitting. In this paper we sketch an alternative strategy. Our analysis commences with the investigation of the free Lie algebra generated by differentiation and by multiplication with the interaction potential: it turns out that this algebra possesses a structure which renders it amenable to a very effective form of asymptotic splitting: exponential splitting where consecutive terms are scaled by increasing powers of the small parameter. This leads to methods which attain high spatial and temporal accuracy and whose cost scales as \({\mathcal {O}}\!\left( M\log M\right) \) , where \(M\) is the number of degrees of freedom in the discretisation.     

Logarithmic Bounds on Sobolev Norms for Time Dependent Linear Schrödinger Equations

We consider the Aharonov–Bohm effect for the Schrödinger operator H = (?i? _x  ? A(x))² + V(x) and the related inverse problem in an exterior domain Ω in R ² with Dirichlet boundary condition. We study the structure and asymptotics of generalized eigenfunctions and show that the scattering operator determines the domain Ω and H up to gauge equivalence under the equal flux condition. We also show that the flux is determined by the scattering operator if the obstacle Ω ^c is convex.     

Small Data Scattering for the Nonlinear Schrödinger Equation on Product Spaces

We consider the cubic nonlinear Schrödinger equation, posed on ? ⁿ  × M, where M is a compact Riemannian manifold and n ≥ 2. We prove that under a suitable smallness in Sobolev spaces condition on the data there exists a unique global solution which scatters to a free solution for large times.     

Stability Estimates for Coefficients of Magnetic Schrödinger Equation from Full and Partial Boundary Measurements

In this paper we establish a log log-type estimate which shows that in dimension n ≥ 3 the magnetic field and the electric potential of the magnetic Schrödinger equation depends stably on the Dirichlet to Neumann (DN) map even when the boundary measurement is taken only on a subset that is slightly larger than half of the boundary ?Ω – a notion made more precise later. Furthermore, we prove that in the case when the measurement is taken on all of ?Ω one can establish a better estimate that is of log-type.     

Remarks on Non-Linear Schrödinger Equation with Magnetic Fields

We study the nonlinear Schrödinger equation with time-depending magnetic field without smallness assumption at infinity. We obtain some results on the Cauchy problem, WKB asymptotics and instability.     

Ground State Solutions for a Quasilinear Schrödinger Equation

In this paper, we study the following quasilinear Schrödinger equation of the form

$$\begin{aligned} -\Delta u+V(x)u-\Delta (u^{2})u= g(x,u),~~~ x\in \mathbb {R}^N \end{aligned}$$

where V and g are 1-periodic in \(x_{1},\ldots ,x_{N}\), and g is a superlinear but subcritical growth as \(|u|\rightarrow \infty \). We develop a more direct and simpler approach to prove the existence of ground state solutions.

     

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.