Related papers: Strichartz estimates for the metaplectic represent…
The discrete Schr\"odinger equation on a two-dimensional honeycomb lattice is a fundamental tight-binding approximation model that describes the propagation of waves on graphene. For free evolution, we first show that the degenerate…
The objective of this paper is to report on recent progress on Strichartz estimates for the Schr\"odinger equation and to present the state-of-the-art. These estimates have been obtained in Lebesgue spaces, Sobolev spaces and, recently, in…
In this work we study symplectic unitary representations for the Galilei group. As a consequence the Schr\"odinger equation is derived in phase space. The formalism is based on the non-commutative structure of the star-product, and using…
We look for the optimal range of Lebesque exponents for which inhomogeneous Strichartz estimates are valid. We show that it is larger than the one given by admissible exponents for homogeneous estimates. We prove inhomogeneous estimates…
We prove Strichartz estimates (both regular and reversed) for a scattering state to the wave equation with a charge transfer Hamiltonian in $\mathbb{R}^{3}$: \[ \partial_{tt}u-\Delta u+\sum_{j=1}^{m}V_{j}\left(x-\vec{v}_{j}t\right)u=0. \]…
We prove modified Strichartz estimates on the one-dimensional torus, that are adapted to a fourth-order dispersion relation, and use them to show global well-posedness of nonlinear fourth-order Schr\"odinger equations. This extends the (low…
The classical Strichartz estimates for the free Schr\"odinger propagator have recently been substantially generalised to estimates of the form \[ \bigg\|\sum_j\lambda_j|e^{it\Delta}f_j|^2\bigg\|_{L^p_tL^q_x}\lesssim\|\lambda\|_{\ell^\alpha}…
We study the dispersive properties of the linear Schr\"odinger equation with a time-dependent potential $V(t,x)$. We show that an appropriate integrability condition in space and time on $V$, i.e. the boundedness of a suitable…
We study local in time Strichartz estimates for the Schroedinger equation associated to long range perturbations of the flat Laplacian on the euclidean space. We prove that in such a geometric situation, outside of a large ball centered at…
Existing algorithms for subgroup discovery with numerical targets do not optimize the error or target variable dispersion of the groups they find. This often leads to unreliable or inconsistent statements about the data, rendering practical…
We consider the iteration of a unitary operator on a separable Hilbert space and study the spreading rates of the associated discrete-time dynamical system relative to a given orthonormal basis. We prove lower bounds for the transport…
We prove Strichartz estimates on any compact connected simple Lie group. In the diagonal case of Bourgain's exponents $p=q,$ we provide a new regularity order $s_{0}^{R}(p)$ in the sense that our (reverse) Strichartz estimates are valid…
In this note, we prove Strichartz estimates for scattering states of scalar charge transfer models in $\mathbb{R}^{3}$. Based on the idea of the proof of Strichartz estimates which follows \cite{CM,RSS}, we also show the energy of the whole…
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…
We obtain necessary and sufficient conditions for the admissible vectors of a new unitary non irreducible representation $U$. The group $G$ is an arbitrary semidirect product whose normal factor $A$ is abelian and whose homogeneous factor…
We establish dispersive estimates and local decay estimates for the time evolution of non-self-adjoint matrix Schr\"odinger operators with threshold resonances in one space dimension. In particular, we show that the decay rates in the…
We study the long time behavior of radial solutions to nonlinear Schr\"{o}dinger equations on hyperbolic space. We show that the usual distinction between short range and long range nonlinearity is modified: the geometry of the hyperbolic…
In this paper we prove the local and global well-posedness of the time fractional abstract Schr\"odinger type evolution equation on the Hilbert space and as an application, we prove the local and global well-posedness of the fractional…
This paper continues the analysis of Schr\"odinger type equations with distributional coefficients initiated by the authors in [3]. Here we consider coefficients that are tempered distributions with respect to the space variable and are…
In this paper, we study time decay estimates for the Schr\"odinger propagator on the product cone $(X,g)$, where $X=C(\rho \mathbb{S}^{n-1})=(0,\infty)\times \rho\mathbb{S}^{n-1}$. We prove that the usual dispersive estimate holds when the…