相关论文: High frequency dispersive estimates in dimension t…
We prove better Strichartz type estimates than expected from the (optimal) dispersion we obtained in our earlier work on a 2d convex model. This follows from taking full advantage of the space-time localization of caustics in the parametrix…
We present a method for two-scale model derivation of the periodic homogenization of the one-dimensional wave equation in a bounded domain. It allows for analyzing the oscillations occurring on both microscopic and macroscopic scales. The…
We prove L^1 --> L^\infty estimates for linear Schroedinger equations in dimensions one and three. The potentials are only required to satisfy some mild decay assumptions. No regularity on the potentials is assumed.
We prove dispersive bounds for fractional Schr\"odinger operators on $\mathbb R^n$ of the form $H=(-\Delta)^{\alpha}+V$ with $V$ a real-valued, decaying potential and $\alpha \notin\mathbb N$. We derive pointwise bounds on the resolvent…
We prove global existence and scattering for a class of quadratic Schrodinger equations in dimension 2. The proof relies on the idea of space-time resonance.
This paper employs layer potential techniques to investigate wave scattering in two-dimensional elastic media exhibiting high contrasts in both Lam\'{e} parameters and density. Our contributions are fourfold. First, we construct an…
The specific character of longitudinal collective electromagnetic oscillations in a layered conductor with the quasi-two-dimensional electron energy spectrum has been analyzed.
In this note we provide dispersive estimates for Fourier integrals with parameter-dependent phase functions in terms of geometric quantities of associated families of Fresnel surfaces. The results are based on a multi-dimensional van der…
High-dimensional group inference is an essential part of statistical methods for analysing complex data sets, including hierarchical testing, tests of interaction, detection of heterogeneous treatment effects and inference for local…
We derive dispersion estimates for solutions of a one-dimensional discrete Dirac equations with a potential. In particular, we improve our previous result, weakening the conditions on the potential. To this end we also provide new results…
Motivated by analogue models of classical and quantum field theory in curved spacetimes and their recent experimental realizations, we consider wave scattering processes of dispersive fields exhibiting two extra degrees of freedom. In…
We prove optimal dispersive estimates for the wave group $e^{it\sqrt{-\Delta+V}}$ for a class of real-valued potentials $V\in C^{(n-3)/2}({\bf R}^n)$, $4\le n\le 7$, such that $\partial_x^\alpha V(x)=O(|x|^{-\delta})$ for $|x|>1$, where…
We study finite dimensional representations of the projective modular group. Various explicit dimension formulas are given.
We show that for a one-dimensional Schr\"odinger operator with a potential whose first moment is integrable the scattering matrix is in the unital Wiener algebra of functions with integrable Fourier transforms. Then we use this to derive…
In this article we shall go over recent work in proving dispersive and Strichartz estimates for the Dirichlet-wave equation. We shall discuss applications to existence questions outside of obstacles and discuss open problems.
We continue our study of scattering theory and dispersive properties for one-dimensional charge transfer models, namely linear Schr\"odinger equations with multiple moving potentials. By the discovery of a refined structure of the…
We derive explicit, uniform, a priori interior Hessian and gradient estimates for special Lagrangian equations of all phases in dimension two.
The supersymmetric approach in the form of second order intertwining relations is used to prove the exact solvability of two-dimensional Schrodinger equation with generalized two-dimensional Morse potential for $a_0=-1/2$. This…
The density stratification in an incompressible fluid is responsible for the propagation of internal waves. In domains with topography, these waves exhibit interesting features. In particular, numerical and lab experiments show that, in two…
We prove global smoothing and Strichartz estimates for the Schroedinger, wave, Klein-Gordon equations and for the massless and massive Dirac systems, perturbed with singular electromagnetic potentials. We impose a smallness condition on the…