Related papers: Counterexamples to Strichartz estimates for the ma…

In space dimension $n\geq3$, we consider the electromagnetic Schr\"odinger Hamiltonian $H=(\nabla-iA(x))^2+V$ and the corresponding Helmholtz equation (\nabla-iA(x))^2u+u+V(x)u=f\quad \text{in}\quad \mathbb{R}^n, where the magnetic and…

In this paper we prove that Schr\"{o}dinger's equation with a Hamiltonian of the form $H=-\Delta+i(A \nabla + \nabla A) + V$, which includes a magnetic potential $A$, has the same dispersive and solution decay properties as the free…

We prove Strichartz estimates for the Schroedinger equation with an electromagnetic potential, in dimension $n\geq3$. The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition,…

We study the Strichartz estimates for the magnetic Schr\"odinger equation in dimension $n\geq3$. More specifically, for all Schr\"odinger admissible pairs $(r,q)$, we establish the estimate $$ \|e^{itH}f\|_{L^{q}_{t}(\mathbb{R};…

We prove Strichartz estimates for the Schr\"odinger equation in $\mathbb R^n$, $n\geq 3$, with a Hamiltonian $H = -\Delta + \mu$. The perturbation $\mu$ is a compactly supported measure in $\mathbb R^n$ with dimension $\alpha >…

In space dimension $n\geq3$, we consider the electromagnetic Schr\"odinger Hamiltonian $H=(\nabla-iA(x))^2-V$ and the corresponding Helmholtz equation $(\nabla-iA(x))^2u+u-V(x)u=f \in \mathbb{R}^n$. We extend the well known $L^p$-$L^q$…

We prove Strichartz estimates for the absolutely continuous evolution of a Schr\"odinger operator $H = (i\nabla + A)^2 + V$ in $\R^n$, $n > 2$. Both the magnetic and electric potentials are time-independent and satisfy pointwise polynomial…

We prove Strichartz estimates for the Schr\"odinger equation with scaling-critical electromagnetic potentials in dimensions $n\geq3$. The decay assumption on the magnetic potentials is critical, including the case of the Coulomb potential.…

In dimension $n>3$ we show the existence of a compactly supported potential in the differentiability class $C^\alpha$, $\alpha < \frac{n-3}2$, for which the solutions to the linear Schr\"odinger equation in $\R^n$, $$ -i\partial_t u = -…

In each dimension n >= 2, we construct a class of nonnegative potentials that are homogeneous of order -sigma, chosen from the range 0 < sigma < 2, and for which the perturbed Schrodinger equation does not satisfy global in time Strichartz…

We consider the semilinear electromagnetic Schr\"{o}dinger equation (-i\nabla+A(x))^{2}u + V(x)u = |u|^{2^{\ast}-2}u, u\in D_{A,0}^{1,2}(\Omega,\mathbb{C}), where $\Omega=(\mathbb{R}^{m}\smallsetminus{0})\times\mathbb{R}^{N-m}$ with $2\leq…

We show that the time evolution of the operator $H = -\Delta + i(A \cdot \nabla + \nabla \cdot A) + V$ in R^3 satisfies Strichartz and smoothing estimates under suitable smoothness and decay assumptions on A and V but without any smallness…

We consider magnetic Schr\"odinger equations with sublinear magnetic potentials and subquadratic electric potentials on $\mathbb{R}^{d}$, as well as generalizations thereof. We obtain new results on the global well-posedness of the Cauchy…

In this paper, we combine the argument of [12] and [27] to prove the maximal estimates for fractional Schr\"odinger equations $(i\partial_t+\mathcal{L}_{\mathbf{A}}^{\frac\alpha 2})u=0$ in the purely magnetic fields which includes the…

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…

We establish an interaction Morawetz estimate for the magnetic Schr\"odinger equation for $n\geq 3$ under certain smallness conditions on the gauge potentials, but with almost optimal decay. As an application, we prove global wellposedness…

For Schr\"odinger equations with a class of slowly decaying repulsive potentials, we show that the solution satisfies global-in-time Strichartz estimates for any admissible pairs. Our admissible class of potentials includes the positive…

We prove some uniform in $\epsilon$ a priori estimates for solutions of the equation $$(\nabla-iA)^2u-V(x)u+(\lambda\pm i\epsilon)u=f, \lambda\geq0, \epsilon\neq0.$$ The estimates are obtained in terms of Morrey-Campanato norms, and can be…

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 purpose of this paper is to study the validity of global-in-time Strichartz estimates for the Schr\"odinger equation on $\mathbb{R}^n$, $n\ge3$, with the negative inverse-square potential $-\sigma|x|^{-2}$ in the critical case…