Related papers: Uniform Sobolev estimates in $\mathbb{R}^{n}$ invo…
Recent (scale-free) quantitative unique continuation estimates for spectral subspaces of Schr\"odinger operators are extended to allow singular potentials such as certain $L^p$-functions. The proof is based on accordingly adapted Carleman…
The Riemann-Hilbert approach is extended to discuss the well-posedness of the nonlinear Schr\"odinger-Gerdjikov-Ivanon equation. The Lipschitz continuity of potential in $H^{2}(\mathbb{R})\cap H^{1,1}(\mathbb{R})$ to scattering data is…
A central result of Sturm-Liouville theory (also called the Sturm-Hurwitz Theorem) states that if $\phi_k$ is a sequence of eigenfunctions of a second order differential operator on the interval $I \subset \mathbb{R}$, then any linear…
Based on a fundamental identity for stochastic hyperbolic-like operators, we derive in this paper a global Carleman estimate (with singular weight function) for stochastic wave equations. This leads to an observability estimate for…
We consider elliptic second order partial differential operators with Lipschitz continuous leading order coefficients on finite cubes and the whole Euclidean space. We prove quantitative sampling and equidistribution theorems for…
We prove a Lieb-Thirring type inequality for potentials such that the associated Schr\"{o}dinger operator has a pure discrete spectrum made of an unbounded sequence of eigenvalues. This inequality is equivalent to a generalized…
We consider the nonlinear Schr\"odinger equations with a potential on $\mathbb T^d$. For almost all potentials, we show the almost global stability in very high Sobolev norms. We apply an iteration of the Birkhoff normal form, as in the…
We prove regularity estimates in weighted Sobolev spaces for the $L^2$-eigenfunctions of Schr\"odinger type operators whose potentials have inverse square singularities and uniform radial limits at infinity. In particular, the usual…
We prove and apply two theorems: First, a quantitative, scale-free unique continuation estimate for functions in a spectral subspace of a Schr\"odinger operator on a bounded or unbounded domain, second, a perturbation and lifting estimate…
Bound-state solutions of the singular harmonic oscillator and singular Coulomb potentials in arbitrary dimensions are generated in a simple way from the solutions of the one-dimensional generalized Morse potential. The nonsingular harmonic…
We prove uniform Sobolev estimates $||u||_{L^{p'}} \leq C ||(\Delta-\alpha)u||_{L^{p}}$, where $p=2n/(n+2), p'=2n/(n-2)$, for the Laplacian $\Delta$ on non-trapping asymptotically conic manifolds of dimension $n$. Here C is independent of…
This paper proves a genericity conjecture by Goldstein, Schlag, and Voda[Invent. Math.\textbf{217}(2019)] for multi-frequency quasiperiodic Schr\"{o}dinger operators. Specifically, we show that for almost all coefficients of real…
We obtain quasimode, eigenfunction and spectral projection bounds for Schr\"odinger operators, $H_V=-\Delta_g+V(x)$, on compact Riemannian manifolds $(M,g)$ of dimension $n\ge2$, which extend the results of the third author~\cite{sogge88}…
We prove Strichartz estimates on general flat d-torus for arbitrary d. Using these estimates, we prove local wellposedness for the cubic nonlinear Schr\"odinger equations in appropriate Sobolev spaces. In dimensions 2 and 3, we prove…
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 inverse Strichartz theorems at $L^2$ regularity for a family of Schr\"{o}dinger evolutions in one space dimension. Prior results rely on spacetime Fourier analysis and are limited to the translation-invariant equation $i\partial_t…
We consider abstract non-negative self-adjoint operators on $L^2(X)$ which satisfy the finite speed propagation property for the corresponding wave equation. For such operators we introduce a restriction type condition which in the case of…
We establish necessary and sufficient conditions for complex potentials in the Schr\"odinger equation to enable spectral singularities (SSs) and show that such potentials have the universal form $U(x) = -w^2(x) - iw_x(x) + k_0^2$, where…
In the first part of the paper we continue the study of solutions to Schr\"odinger equations with a time singularity in the dispersive relation and in the periodic setting. In the second we show that if the Schr\"odinger operator involves a…
In this paper, we study the convergence of the spectral embeddings obtained from the leading eigenvectors of certain similarity matrices to their population counterparts. We opt to study this convergence in a uniform (instead of average)…