Related papers: A Short Course on One-Dimensional Random Schr\"odi…
In this paper we solve a long standing open problem for Random Schr\"odinger operators on $L^2(\mathbb{R}^d)$ with i.i.d single site random potentials. We allow a large class of free operators, including magnetic potential, however our…
We prove maximal regularity results in H\"older and Zygmund spaces for linear stationary and evolution equations driven by a large class of differential and pseudo-differential operators L, both in finite and in infinite dimension. The…
Time-frequency analysis have played a crucial role in the development of localization operators in the last twenty years. We present its applications to the study of boundedness and Schatten Class property for such operators. In particular,…
We study a particular class of families of multi-dimensional lattice Schr\"o\-dinger operators with deterministic (including quasi-periodic) potentials generated by the "hull" given by an orthogonal series over the Haar wavelet basis on the…
We consider uniformly subelliptic operators on certain unimodular Lie groups of polynomial growth. It was shown by Saloff-Coste and Stroock that classical results of De Giorgi, Nash, Moser, Aronson extend to this setting. It was then…
We extend earlier work [Phys.Rev.Lett. 84, 3740 (2000)] on the statistical mechanics of the cubic one-dimensional discrete nonlinear Schrodinger (DNLS) equation to a more general class of models, including higher dimensionalities and…
We consider unitary analogs of $1-$dimensional Anderson models on $l^2(\Z)$ defined by the product $U_\omega=D_\omega S$ where $S$ is a deterministic unitary and $D_\omega$ is a diagonal matrix of i.i.d. random phases. The operator $S$ is…
In this paper, we study the existence of the random approximations and fixed points for random almost lower semicontinuous operators defined on finite dimensional Banach spaces, which in addition, are condensing or 1-set-contractive. Our…
We prove the universality of sharp arithmetic localization for all one-dimensional quasiperiodic Schr\"odinger operators with anti-Lipschitz monotone potentials.
In this work, we introduce and study nonlinear Schr\"odinger equations (NLS) with anisotropic dispersion, where the standard Laplacian acts on the Euclidean variable \(x \in \mathbb{R}^d\), and an Ornstein-Uhlenbeck ($\mathcal{OU}$)…
In this paper we provide an extension theorem for fractional powers of some pseudo-differential operators $P(D)$. These extensions yields realization of the fractional powers of some pseudo-differential operators in the spirit of Caffarelli…
This is a survey of recent progress on the irreducibility of Fermi varieties, rigidity results and embedded eigenvalue problems of discrete periodic Schr\"odinger operators.
In joint work with Adam Black and Reuben Drogin, we develop a new approach to understanding the diffusive limit of the random Schrodinger equation based on ideas taken from random matrix theory. These lecture notes present the main ideas…
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 define a class of pseudo-ergodic non-self-adjoint Schr\"odinger operators acting in spaces $l^2(X)$ and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a non-self-adjoint Anderson…
Recent sampling theorems allow for the recovery of operators with bandlimited Kohn-Nirenberg symbols from their response to a single discretely supported identifier signal. The available results are inherently non-local. For example, we…
Local scale invariance for lattice models is studied using new realizations of the Schr\"odinger algebra. The two-point function is calculated and it turns out that the result can be reproduced from exact two-point correlation functions…
The aim of this article is to introduce Vogel's localization theorem for classes of D-complexes: this generalization of Waldhausen's localization theorem is especially useful and powerful in that it gives an explicit and computable…
This lecture notes cover a Part III (first year graduate) course that was given at Cambridge University over several years on pseudo-differential operators. The calculus on manifolds is developed and applied to prove propagation of…
This paper investigates the localization properties of solutions to the semi-classical Schr\"odinger equation on closed Riemann surfaces. Unlike classical studies that assume a smooth potential, our work addresses the challenges arising…