Operator kernel estimates for functions of generalized Schrödinger operators

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of mathematical physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved Combes-Thomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than any polynomial.

     

Unique continuation for the Schrödinger equation with gradient vector potentials

We obtain unique continuation results for Schrödinger equations with time dependent gradient vector potentials. This result with an appropriate modification also yields unique continuation properties for solutions of certain nonlinear Schrödinger equations.

     

On the singular spectrum of Schrödinger operators with decaying potential

The relation between Hausdorff dimension of the singular spectrum of a Schrödinger operator and the decay of its potential has been extensively studied in many papers. In this work, we address similar questions from a different point of view. Our approach relies on the study of the so-called Krein systems. For Schrödinger operators, we show that some bounds on the singular spectrum, obtained recently by Remling and Christ-Kiselev, are optimal.

     

锥中与稳态的薛定谔算子相关的广义Martin函数无穷远处的控制

本文研究了稳态的薛定谔算子的Dirichlet问题和Martin函数的边界行为.利用广义Martin表示和稳态的薛定谔算子对应的常微分方程基本解,在具有光滑边界的锥形区域中获得了与稳态的薛定谔算子相关的广义Martin函数无穷远处广义调和控制的一些刻画,推广了拉普拉斯算子情形的结果.     

Solving Nonlinear p-Adic Pseudo-differential Equations: Combining the Wavelet Basis with the Schauder Fixed Point Theorem

We consider the pointwise convergence problem for the solution of Schrödinger-type equations along directions determined by a given compact subset of the real line. This problem contains Carleson’s problem as the simplest case and was studied in general by Cho et al. We extend their result from the case of the classical Schrödinger equation to a class of equations which includes the fractional Schrödinger equations. To achieve this, we significantly simplify their proof by completely avoiding a time localization argument.     

Sub-exponential decay of operator kernels for functions of generalized Schrödinger operators

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators. We prove sub-exponential decay for functions in Gevrey classes and exponential decay for real analytic functions.

     

Inhomogeneous Strichartz estimates for the Schrödinger equation

We study Strichartz estimates for the solution of the Cauchy problem associated with the inhomogeneous free Schrödinger equation in the case when the inital data is equal to zero, proving some new estimates for certain exponents and giving counterexamples for some others.

     

Cutoff resolvent estimates and the semilinear Schrödinger equation

This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schrödinger equation. If the resolvent estimate has a loss when compared to the optimal, non-trapping estimate, there is a corresponding loss in regularity in the local smoothing estimate. As an application, we apply well-known techniques to obtain well-posedness results for the semi-linear Schrödinger equation.

     

Local Nonautonomous Schrödinger Flows on Kähler Manifolds

In this paper, we prove that the nonautonomous Schrödinger flow from a compact Riemannian manifold into a Kähler manifold admits a local solution     

Finite propagation speed and kernel estimates for Schrödinger operators

We point out finite propagation speed phenomena for discrete and continuous Schrödinger operators and discuss various types of kernel estimates from this point of view.

     

Lorentz space extension of Strichartz estimates

In this paper, Strichartz estimates for the solution of the Schrödinger evolution equation are considered on a mixed normed space with Lorentz norm with respect to the time variable.

     

Asymptotics of eigenvalue clusters for Schrödinger operators on the Sierpinski gasket

In this note we investigate the asymptotic behavior of spectra of Schrödinger operators with continuous potential on the Sierpinski gasket . In particular, using the existence of localized eigenfunctions for the Laplacian on we show that the eigenvalues of the Schrödinger operator break into clusters around certain eigenvalues of the Laplacian. Moreover, we prove that the characteristic measure of these clusters converges to a measure. Results similar to ours were first observed by A. Weinstein and V. Guillemin for Schrödinger operators on compact Riemannian manifolds.

     

On quadratic derivative Schrödinger equations in one space dimension

We consider the Schrödinger equation with derivative perturbation terms in one space dimension. For the linear equation, we show that the standard Strichartz estimates hold under specific smallness requirements on the potential. As an application, we establish existence of local solutions for quadratic derivative Schrödinger equations in one space dimension with small and rough Cauchy data.

     

Hydrodynamic approach to constructing solutions of the nonlinear Schrödinger equation in the critical case

Proceeding from the hydrodynamic approach, we construct exact solutions to the nonlinear Schrödinger equation with special properties. The solutions describe collapse, in finite time, and scattering, over infinite time, of wave packets. They generalize known blow-up solutions based on the ``ground state'.

     

Asymptotic Dirichlet problem for the Schrödinger operator via rough isometry

We pose and solve the asymptotic Dirichlet problem for the Schrödinger operator via rough isometries on a certain class of Riemannian manifolds. With suitable potentials, we give the solvability of the problem for a naturally defined class of data functions.

     

Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators

New unique characterization results for the potential in connection with Schrödinger operators on and on the half-line are proven in terms of appropriate Krein spectral shift functions. Particular results obtained include a generalization of a well-known uniqueness theorem of Borg and Marchenko for Schrödinger operators on the half-line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for Schrödinger operators with confining potentials on the entire real line.

     

一类Schrödinger型算子在关于非负位势的变指数Morrey空间上的有界性

本文考虑了一类Schrödinger型算子和其交换子的有界性问题.利用其在Lp空有界性间上的,获得了一类Schrödinger型算子和其交换子在变指数Morrey空间上的有界性.     

Semi-classical behavior of the spectral function

We study the semi-classical behavior of the spectral function of the Schrödinger operator with short range potential. We prove that the spectral function is a semi-classical Fourier integral operator quantizing the forward and backward Hamiltonian flow relations of the system. Under a certain geometric condition we explicitly compute the phase in an oscillatory integral representation of the spectral function.

     

Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity

Foundations of Computational Mathematics - We present a new filtered low-regularity Fourier integrator for the cubic nonlinear Schrödinger equation based on recent time discretization and...     

Beurling-Nevanlinna inequality for subfunctions of the stationary Schrödinger operator

The classical Beurling-Nevanlinna upper bound for subharmonic functions is extended to subsolutions of the stationary Schrödinger equation.