Related papers: Improved local smoothing estimate for the fraction…
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…
We study the cubic nonlinear fractional Schr\"odinger equation with L\'evy indices $\frac{4}{3}<\alpha< 2$ posed on the half-line. More precisely, we define the notion of a solution for this model and we obtain a result of…
We establish global two-sided heat kernel estimates (for full time and space) of the Schr\"odinger operator $-\frac{1}{2}\Delta+V$ on $\R^d$, where the potential $V(x)$ is locally bounded and behaves like $c|x|^{-\alpha}$ near infinity with…
We prove a sharp resolvent estimate in scale invariant norms of Amgon--H\"{o}rmander type for a magnetic Schr\"{o}dinger operator on $\mathbb{R}^{n}$, $n\ge3$\begin{equation*} L=-(\partial+iA)^{2}+V \end{equation*}with large potentials…
We will show that a local space-time estimate implies a global space-time estimate for dispersive operators. In order for this implication we consider a Littlewood-Paley type square function estimate for dispersive operators in a time…
We obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schr\"odinger operator with negative Hardy potential $$\Delta^{\alpha/2} -\lambda |x|^{-\alpha}$$ on $\RR^d$, where…
We prove a local in time smoothing estimate for a magnetic Schrodinger equation with coefficients growing polynomially at spatial infinity. The assumptions on the magnetic field are gauge invariant and involve only the first two…
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…
The present work aims at obtaining estimates for transformation operators for one-dimensional perturbed radial Schr\"odinger operators. It provides more details and suitable extensions to already existing results, that are needed in other…
A local time decay estimate of fractional Schr\"odinger operators with slowly decaying positive potentials are studied. It is shown that its resolvent is smooth near zero and the time propagator has fast local time decay which is very…
We obtain weighted $L^2$ Strichartz estimates for Schr\"odinger equations $i\partial_tu+(-\Delta)^{a/2}u=F(x,t)$, $u(x,0)=f(x)$, of general orders $a>1$ with radial data $f,F$ with respect to the spatial variable $x$, whenever the weight is…
We consider the nonlinear Schr\"odinger equation with dispersion modulated by a (formal) derivative of a time-dependent function with fractional Sobolev regularity of class $W^{\alpha,2}$ for some $\alpha\in (0,1)$. Due to the loss of…
We introduce the Hardy spaces for Fourier integral operators on Riemannian manifolds with bounded geometry. We then use these spaces to obtain improved local smoothing estimates for Fourier integral operators satisfying the cinematic…
The work is devoted to Dirichlet problem for sub-quintic semi-linear wave equation with damping damping term of the form $(-\Delta)^\alpha\partial_t u$, $\alpha\in(0,\frac{1}{2})$, in bounded smooth domains of $\Bbb R^3$. It appears that to…
In this note we prove the estimate $M^{\sharp}_{0,s}(Tf)(x) \le c\,M_\gamma f(x)$ for general fractional type operators $T$, where $M^{\sharp}_{0,s}$ is the local sharp maximal function and $M_\gamma$ the fractional maximal function, as…
We study the maximal regularity problem for abstract time-fractional Schr\"odinger equations $\partial_t^\alpha(u-u_0) -\mathrm{i} A u=f$, with a fractional derivative $\partial_t^\alpha$ of order $\alpha \in (0,1)$. We assume that $A$ is a…
The paper is concerned with a posteriori estimates for approximations of boundary value problems generated by the spectral fractional Laplace operator. The derivation is based upon the Stinga--Torrea extension, which generalizes the…
We develop a refined Frozen Gaussian approximation (FGA) for the fractional Schr\"odinger equation in the semi-classical regime, where the solution exhibits rapid oscillations as the scaled Planck constant $\varepsilon$ becomes small. Our…
We consider an elliptic equation with the fractional Laplacian operator $(-\Delta)^{\frac{\alpha}{2}}$ in the dissipative term, a singular integral operator ${\bf A}(\cdot)$ in the nonlinear term, and an external source $f$. The key example…
Two-sided estimates for higher order eigenvalues are presented for a class of non-local Schr\"odinger operators by using the jump rate and the growth of the potential. For instance, let $L$ be the generator of a L\'evy process with L\'evy…