Related papers: The space-time estimates for the Schr\"odinger equ…
We consider the Schr\"odinger map initial value problem into the sphere in 2+1 dimensions with smooth, decaying, subthreshold initial data. Assuming an a priori $L^4$ boundedness condition on the solution, we prove that the Schr\"odinger…
In this work, we mainly focus on the energy-supercritical nonlinear Schr\"odinger equation, $$ i\partial_{t}u+\Delta u= \mu|u|^p u, \quad (t,x)\in \mathbb{R}^{d+1}, $$ with $\mu=\pm1$ and $p>\frac4{d-2}$. %In this work, we consider the…
We consider the nonlinear Schr{\"o}dinger equation with a potential, also known as Gross-Pitaevskii equation. By introducing a suitable spectral localization, we prove low regularity error estimates for the time discretization corresponding…
The time-global existence of solutions to a system of stochastic Schr\"odinger equations with multiplicative noise and the quadratic nonlinear terms are discussed in this paper. The same system in the deterministic treatment was studied in…
We prove smoothing estimates for Schr\"odinger equations $i\partial_t \phi+\partial_x (a(x) \partial_x \phi) =0$ with $a(x)\in \mathrm{BV}$, the space of functions with bounded total variation, real, positive and bounded from below. We then…
We study the computability of global solutions to linear Schr\"odinger equations with magnetic fields and the Hartree equation on $\mathbb R^3$. We show that the solution can always be globally computed with error control on the entire…
The present paper is concerned with Schr\"odinger equations with variable coefficients and unbounded electromagnetic potentials, where the kinetic energy part is a long-range perturbation of the flat Laplacian and the electric (resp.…
We prove L^1 --> L^\infty estimates for the linear Schroedinger equation in three dimensions. The potential is assumed to belong to certain L^p spaces, but no pointwise decay estimates and no additional regularity is required.
In this paper we prove some multi-linear Strichartz estimates for solutions to the linear Schr\"odinger equations on torus $\T^n$. Then we apply it to get some local well-posed results for nonlinear Schr\"odinger equation in critical…
Results of P. Sj\"olin and F. Soria on the Schr\"odinger maximal operator with complex-valued time are improved by determining up to the endpoint the sharp $s \geq 0$ for which boundedness from the Sobolev space $H^s(\mathbb{R})$ into…
In this paper we give an estimate for the solution to the Schr\"odinger equation with sub-quadratic potentials in modulation spaces by the norm of the initial functions in Wiener-Amalgum spaces.
We prove optimal high-frequency resolvent estimates for perturbations by large magnetic and electric potentials
We develop a spacetime neural network method with second order optimization for solving quantum dynamics from the high dimensional Schr\"{o}dinger equation. In contrast to the standard iterative first order optimization and the…
In this paper, we establish refined Strichartz estimates for higher-order Schr\"odinger equations with initial data exhibiting partial regularity. By partial regularity, we mean that the initial data are not required to have full Sobolev…
We discuss the global existence of solutions to a system of stochastic Schr\"odinger equations with multiplicative noise. Our setting of the quadratic nonlinear terms in dimension 4 is $L^2$-critical. We treat the solutions under the ground…
We prove dispersive estimates for solutions to the Schrodinger equation with a real-valued potential $V\in L^\infty({\bf R}^n)$, $n\ge 4$, satisfying $V(x)=O(|x|^{-(n+2)/2-\epsilon})$, $|x|>1$, $\epsilon>0$.
We present a unified approach for obtaining global a priori estimates for solutions of nonlinear defocusing Schr\"odinger equations with defocusing nonlinearities. The estimates are produced by contracting the local momentum conservation…
We consider a family of surfaces of revolution, each with a single periodic geodesic which is degenerately unstable. We prove a local smoothing estimate for solutions to the linear Schr\"odinger equation with a loss that depends on the…
We consider a nonlinear Schr{\"o}dinger equation set in the whole space with a single power of interaction and an external source. We first establish existence and uniqueness of the solutions and then show, in low space dimension, that the…
In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schr\"odinger equations for the spatial dimension $d=2$ and $3$. This system was introduced by M. Colin and T. Colin (2004). The first…