Related papers: A global space-time estimate for dispersive operat…
We consider the uniform resolvent and orthonormal Strichartz estimates for the Schr\"odinger operator. First we prove the Keel-Tao type theorem for the orthonormal Strichartz estimates, which means that the dispersive estimates yield the…
A method for deriving the Schrodinger equation for Lagrangian path integral with scaling of local time is given.
The present paper is dedicated to the proof of dispersive estimates on stratified Lie groups of step 2, for the linear Schr\"odinger equation involving a sublaplacian. It turns out that the propagator behaves like a wave operator on a space…
This paper is a survey article of results and arguments from several of authors' papers, and it describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on ideas of comparison…
In $L_2 (\mathbb{R}^d; \mathbb{C}^n)$, we consider a selfadjoint matrix strongly elliptic second order differential operator $\mathcal{A}_\varepsilon$ with periodic coefficients depending on $\mathbf{x}/\varepsilon$. We find approximations…
The classical $L^2$ estimate for the $\overline{\partial}$ operators is a basic tool in complex analysis of several variables. Naturally, it is expected to extend this estimate to infinite dimensional complex analysis, but this is a…
The nonrelativistic Schroedinger equation for motion of a structureless particle in four-dimensional space-time entails a well-known expression for the conserved four-vector field of local probability density and current that are associated…
In this work, we prove global well-posedness and scattering for systems of quadratic nonlinear Schr\"odinger equations in the critical three-dimensional case, for small, localized data. For the terms corresponding to the nonlinearity…
We study the global existence and decay estimates for nonlinear wave equations with the space-time dependent dissipative term in an exterior domain. The linear dissipative effect may vanish in a compact space region. Moreover the nonlinear…
We investigate $L^1(\mathbb R^n)\to L^\infty(\mathbb R^n)$ dispersive estimates for the Schr\"odinger operator $H=-\Delta+V$ when there is an eigenvalue at zero energy in even dimensions $n\geq 6$. In particular, we show that if there is an…
With the rapid advances of data acquisition techniques, spatio-temporal data are becoming increasingly abundant in a diverse array of disciplines. Here we develop spatio-temporal regression methodology for analyzing large amounts of…
The translation of an operator is defined by using conjugation with time-frequency shifts. Thus, one can define $\Lambda$-shift-invariant subspaces of Hilbert-Schmidt operators, finitely generated, with respect to a lattice $\Lambda$ in…
We establish the existence of Bogoliubov's local scattering operators for P(\phi)_2 models of constructive quantum field theory in a nonperturbative way. To this end, we use the technique of evolution semigroups to prove a new result on…
Strichartz estimates for a time-decaying harmonic oscillator were proven with some assumptions of coefficients for the time-decaying harmonic potentials. The main results of this paper are to remove these assumptions and to enable us to…
The estimation of covariance operators of spatio-temporal data is in many applications only computationally feasible under simplifying assumptions, such as separability of the covariance into strictly temporal and spatial factors.Powerful…
This paper is devoted to the error analysis of a time-spectral algorithm for fractional diffusion problems of order $\alpha$ ($0 < \alpha < 1$). The solution regularity in the Sobolev space is revisited, and new regularity results in the…
We introduce computable projection operators onto piecewise polynomial spaces, defined via sampling and discrete least-squares polynomial approximations. The resulting mappings exhibit (almost) optimal approximation properties in $L^2$ and…
In this paper, we consider the dispersive estimates for Schr\"odinger operators with Coulomb-like decaying potentials, such as $V(x)=-c|x|^{-\mu}$ for $|x|\gg 1$ with $0<\mu<2$, in one dimension. As an application, we establish both 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.
This paper is devoted to study the dispersive properties of the linear Klein-Gordon and wave equations on a class of locally symmetric spaces. As a consequence, we obtain the Strichartz estimate and prove global well-posedness results for…