相关论文: Uncertainty Principles, Spectral Localization, and…

In this paper, we prove pure point spectrum for a large class of Schr\"odinger operators over circle maps with conditions on the rotation number going beyond the Diophantine. More specifically, we develop the scheme to obtain pure point…

We prove eigenvalue bounds for Schr\"odinger operator $-\Delta_g+V$ on compact manifolds with complex potentials $V$. The bounds depend only on an $L^q$-norm of the potential, and they are shown to be optimal, in a certain sense, on the…

Starting from the semi-classical spectrum of Schr\"odinger operators $-h^2\Delta+V$ (on $\mathbb{R}^n$ or on a Riemannian manifold) it is possible to detect critical levels of the potential $V$. Via micro-local methods one can express…

Schr\"odinger operators with periodic (possibly complex-valued) potentials and discrete periodic operators (possibly with complex-valued entries) are considered, and in both cases the computational spectral problem is investigated: namely,…

We consider an inverse spectral problem on a quantum graph associated with the square lattice. Assuming that the potentials on the edges are compactly supported and symmetric, we show that the Dirichlet-to-Neumann map for a boundary value…

Using an extension of the H\"ormander product of distributions, we obtain an intrinsic formulation of one-dimensional Schr\"odinger operators with singular potentials. This formulation is entirely defined in terms of standard {\it Schwartz}…

We generalize the Strichartz estimates for Schr\"odinger operators on compact manifolds of Burq, G\'erard and Tzvetkov [10] by allowing critically singular potentials $V$. Specifically, we show that their $1/p$--loss $L^p_tL^q_x(I\times…

We establish quantitative upper and lower bounds for Schr\"odinger operators with complex potentials that satisfy some weak form of sparsity. Our first result is a quantitative version of an example, due to S.\ Boegli (Comm. Math. Phys.,…

We prove a Hardy inequality for uniformly elliptic operators subject to Dirichlet or mixed boundary conditions on domains $\Omega$ with piecewiese smooth boundary in arbitrary Riemannian Manifolds (M, g). Employing an approach of E.B.…

In this paper, we develop a systematic framework to study the dispersion surfaces of Schr{\"o}dinger operators $ -\Delta + V$, where the potential $V \in C^\infty(\mathbb{R}^n,\mathbb{R})$ is periodic with respect to a lattice $\Lambda…

We develop a sharp boundary trace theory in arbitrary bounded Lipschitz domains which, in contrast to classical results, allows "forbidden" endpoints and permits the consideration of functions exhibiting very limited regularity. This is…

The main objective of this paper is to systematically develop a spectral and scattering theory for selfadjoint Schr\"odinger operators with $\delta$-interactions supported on closed curves in $\mathbb R^3$. We provide bounds for the number…

We introduce compactness classes of Hilbert space operators by grouping together all operators for which the associated singular values decay at a certain speed and establish upper bounds for the norm of the resolvent of operators belonging…

We study spectral approximations of Schr\"odinger operators $T=-\Delta+Q$ with complex potentials on $\Omega=\mathbb{R}^d$, or exterior domains $\Omega\subset \mathbb{R}^d$, by domain truncation. Our weak assumptions cover wide classes of…

We prove a spectral inequality (a specific type of uncertainty relation) for Schr\"odinger operators with confinement potentials, in particular of Shubin-type. The sensor sets are allowed to decay exponentially, where the precise allowed…

It is well known that uniform resolvent estimates imply spectral cluster estimates. We show that the converse is also true in some cases. In particular, the universal spectral cluster estimates of Sogge \cite{MR930395} for the…

We show that, for one-dimensional discrete Schr\"odinger operators, stability of Anderson localization under a class of rank one perturbations implies absence of intervals in spectra. The argument is based on well-known result of Gordon and…

The behaviour of the spectral edges (embedded eigenvalues and resonances) is discussed at the two ends of the continuous spectrum of non-local discrete Schr\"odinger operators with a $\delta$-potential. These operators arise by replacing…

We give a sharp upper bound on the vanishing order of solutions to Schr\"odinger equation, in the case that the potential is of class $\mathcal{C}^1$ on a smooth compact manifold.

This article is a continuation of arXiv:2401.14977. We study the concentration properties of spectral projectors on manifolds, in connection with the uncertainty principle. In arXiv:2401.14977, the second author proved an optimal…