Related papers: Quasimode and Strichartz estimates for time-depend…
Based on the recent work \cite{KKK} for compact potentials, we develop the spectral theory for the one-dimensional discrete Schr\"odinger operator $$ H \phi = (-\De + V)\phi=-(\phi_{n+1} + \phi_{n-1} - 2 \phi_n) + V_n \phi_n. $$ We show…
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…
This paper deals with global dispersive properties of Schr\"odinger equations with real-valued potentials exhibiting critical singularities, where our class of potentials is more general than inverse-square type potentials and includes…
We establish resolvent estimates that extend earlier results to a larger class of electric potentials $V\in L^\infty(\mathbb{R}^d;\mathbb{R})$, $d\ge 3$, and magnetic potentials $b\in L^\infty(\mathbb{R}^d;\mathbb{R}^d)$ such that $V(x),…
Sharp Strichartz estimates are proved for Schr\"odinger and wave equations with Lipschitz coefficients satisfying additional structural assumptions. We use Phillips functional calculus as a substitute for Fourier inversion, which shows how…
We prove sharp Strichartz-type estimates in three dimensions, including some which hold in reverse spacetime norms, for the wave equation with potential. These results are also tied to maximal operator estimates studied by…
We investigate the quantitative unique continuation properties of solutions to second order elliptic equations with singular lower order terms. The main theorem presents a quantification of the strong unique continuation property for…
We derive H\"older regularity estimates for operators associated with a time independent Schr\"odinger operator of the form $-\Delta+V$. The results are obtained by checking a certain condition on the function $T1$. Our general method…
Numerical solving the Schr\"odinger equation with incommensurate potentials presents a great challenge since its solutions could be space-filling quasiperiodic structures without translational symmetry nor decay. In this paper, we propose…
In this article, we first prove quantitative estimates associated to the unique continuation theorems for operators with partially analytic coefficients of Tataru, Robbiano-Zuily and H\"ormander. We provide local stability estimates that…
For Schr\"{o}dinger equations with potentials which grow at most quadratically at spatial infinity, we prove Strichartz estimates in Wiener amalgam spaces. These estimates provide a stronger recovery of local-in-space regularity than the…
We study eigenvalues of non-self-adjoint Schr\"odinger operators on non-trapping asymptotically conic manifolds of dimension $n\ge 3$. Specifically, we are concerned with the following two types of estimates. The first one deals with Keller…
We deal with fixed-time and Strichartz estimates for the Schr\"odinger propagator as an operator on Wiener amalgam spaces. We discuss the sharpness of the known estimates and we provide some new estimates which generalize the classical…
We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space. These inequalities lead to universal bounds on spectral gaps and on moments of eigenvalues lambda_k that are analogous to those known for Schroedinger…
Suppose $L=-\Delta+V$ is a Schr\"odinger operator on $\mathbb{R}^n$ with a potential $V$ belonging to certain reverse H\"older class $RH_\sigma$ with $\sigma\geq n/2$. The main aim of this paper is to provide necessary and sufficient…
The completeness of the quasinormal modes of the wave equation with Poeschl-Teller potential is investigated. A main result is that after a large enough time $t_0$, the solutions of this equation corresponding to $C^{\infty}$-data with…
We eliminate by KAM methods the time dependence in a class of linear differential equations in $\ell^2$ subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator $…
We study the cut-off resolvent of semiclassical Schr{\"o}dinger operators on $\mathbb{R}^d$ with bounded compactly supported potentials $V$. We prove that for real energies $\lambda^2$ in a compact interval in $\mathbb{R}_+$ and for any…
We prove that if a solution of the discrete time-dependent Schr\"odinger equation with bounded real potential decays fast at two distinct times then the solution is trivial. For the free Shr\"odinger operator and for operators with…
We give a sharp upper bound on the vanishing order of solutions to Schrodinger equation with C^1 electric and magnetic potentials on a compact smooth manifold. Our method is based on quantitative Carleman type inequalities developed by…