Related papers: On the Schrodinger equation outside strictly conve…
We prove resolvent estimates for semiclassical operators such as $-h^2 \Delta+V(x)$ in scattering situations. Provided the set of trapped classical trajectories supports a chaotic flow and is sufficiently filamentary, the analytic…
We establish new Strichartz estimates for orthonormal systems on compact Riemannian manifolds in the non-sharp admissible region of exponents, covering wave, Klein-Gordon, and fractional Schr\"odinger equations. Our approach combines the…
We construct a local in time, exponentially decaying solution of the one-dimensional variable coefficient Schrodinger equation by solving a nonstandard boundary value problem. A main ingredient in the proof is a new commutator estimate…
We study the well-posedness of the Cauchy problem with Dirichlet or Neumann boundary conditions associated to an H 1 -critical semilinear wave equation on a smooth bounded 2D domain {\Omega}. First, we prove an appropriate Strichartz type…
We prove observability estimates for the Schr\"odinger equation posed on the equilateral triangle in the plane, under both Neumann and Dirichlet boundary conditions. No geometric control condition is required on the rough localization…
We prove an explicit weighted estimate for the semiclassical Schr\"odinger operator $P = - h^2 \partial^2_x + V(x;h)$ on $L^2(\mathbb{R})$, with $V(x;h)$ a finite signed measure, and where $h >0$ is the semiclassical parameter. The proof is…
We establish global well-posedness and scattering for the cubic Dirac equation for small data in the critical space $H^1(\mathbb{R}^3)$. The main ingredient is obtaining a sharp end-point Strichartz estimate for the Klein-Gordon equation.…
We prove Strichartz estimates with a loss of derivatives for the Schr\"odinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates the on polygon…
We consider Schr\"odinger equation with a non-degenerate metric on the Euclidean space. We study local in time Strichartz estimates for the Schr\"odinger equation without loss of derivatives including the endpoint case. In contrast to the…
Considering stochastic partial differential equations of parabolic type with random coefficients in vector-valued H\"older spaces, we obtain a sharp Schauder estimate. As an application, the existence and uniqueness of solution to the…
We consider the cubic-quintic nonlinear Schr\"odinger equation in two space dimensions. For this model, X. Cheng established scattering for $H^1$ data with mass strictly below that of the ground state for the cubic NLS. Subsequently, R.…
In this paper, we study the defocusing energy-critical nonlinear Schr\"odinger equations $$ i\partial_t u + \Delta u = |u|^{\frac{4}{d-2}} u. $$ When $d=3,4$, we prove the almost sure scattering for the equations with non-radial data in…
In this paper we present a method to study global regularity properties of solutions of large-data critical Schrodinger equations on certain noncompact Riemannian manifolds. We rely on concentration compactness arguments and a global…
In this paper we prove the orthonormal Strichartz estimates for the higher order and fractional Schr\"odinger, wave, Klein-Gordon and Dirac equations with potentials. As in the case of the Schr\"odinger operator, the proofs are based on the…
We consider the semiclassical Schr\"odinger equation on a compact negatively curved surface. For any sequence of initial data microlocalized on the unit cotangent bundle, we look at the quantum evolution (below the Ehrenfest time) under…
In this paper we prove local-in-time Strichartz estimates with loss of derivatives for Schr\"odinger equations with variable coefficients and potentials, under the conditions that the geodesic flow is nontrapping and potentials grow…
We consider the defocusing nonlinear Schr{\"o}dinger equation in several space dimensions, in the presence of an external potential depending on only one space vari-able. This potential is bounded from below, and may grow arbitrarily fast…
In this note we obtain some Strichartz estimates for the Schr\"odinger equation associated to the twisted Laplacian on $\mathbb{C}^{n}\cong \mathbb{R}^{2n}$. The initial data will be considered in suitable Sobolev spaces associated to the…
We prove bilinear estimates for the Schr\"odinger equation on 3D domains, with Dirichlet boundary conditions. On non-trapping domains, they match the $\mathbb{R}^3$ case, while on bounded domains they match the generic boundary less…
We prove almost sure global existence and scattering for the energy-critical nonlinear Schr\"odinger equation with randomized spherically symmetric initial data in $H^s(\mathbb{R}^4)$ with $\frac56<s<1$. We were inspired to consider this…