Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userEigenvalue statistics for Schrodinger operators with random point interactions on $mathbb{R}^d$, $d=1,2,3$
152
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove that the local eigenvalue statistics at energy $E$ in the localization regime for Schrodinger operators with random point interactions on $mathbb{R}^d$, for $d=1,2,3$, is a Poisson point process with the intensity measure given by the density of states at $E$ times the Lebesgue measure. This is one of the first examples of Poisson eigenvalue statistics for the localization regime of multi-dimensional random Schrodinger operators in the continuum. The special structure of resolvent of Schrodinger operators with point interactions facilitates the proof of the Minami estimate for these models.
rate research
Read More
We prove Anderson localization at the internal band-edges for periodic magnetic Schr{o}dinger operators perturbed by random vector potentials of Anderson-type. This is achieved by combining new results on the Lifshitz tails behavior of the integrated density of states for random magnetic Schr{o}dinger operators, thereby providing the initial length-scale estimate, and a Wegner estimate, for such models.
In this paper, we prove a power-law version dynamical localization for a random operator $mathrm{H}_{omega}$ on $mathbb{Z}^d$ with long-range hopping. In breif, for the linear Schrodinger equation $$mathrm{i}partial_{t}u=mathrm{H}_{omega}u, quad u in ell^2(mathbb{Z}^d), $$ the Sobolev norm of the solution with well localized initial state is bounded for any $tgeq 0$.
Let $H_V=-Delta +V$ be a Schrodinger operator on an arbitrary open set $Omega$ of $mathbb R^d$, where $d geq 3$, and $Delta$ is the Dirichlet Laplacian and the potential $V$ belongs to the Kato class on $Omega$. The purpose of this paper is to show $L^p$-boundedness of an operator $varphi(H_V)$ for any rapidly decreasing function $varphi$ on $mathbb R$. $varphi(H_V)$ is defined by the spectral theorem. As a by-product, $L^p$-$L^q$-estimates for $varphi(H_V)$ are also obtained.
We obtain new Faber-Krahn-type inequalities for certain perturbations of the Dirichlet Laplacian on a bounded domain. First, we establish a two- and three-dimensional Faber-Krahn inequality for the Schrodinger operator with point interaction: the optimiser is the ball with the point interaction supported at its centre. Next, we establish three-dimensional Faber-Krahn inequalities for one- and two-body Schrodinger operator with attractive Coulomb interactions, the optimiser being given in terms of Coulomb attraction at the centre of the ball. The proofs of such results are based on symmetric decreasing rearrangement and Steiner rearrangement techniques; in the first model a careful analysis of certain monotonicity properties of the lowest eigenvalue is also needed.
In terms of the minimal $N$-point diameter $D_d(N)$ for $R^d,$ we determine, for a class of continuous real-valued functions $f$ on $[0,+infty],$ the $N$-point $f$-best-packing constant $min{f(|x-y|), :, x,yin R^d}$, where the minimum is taken over point sets of cardinality $N.$ We also show that $$ N^{1/d}Delta_d^{-1/d}-2le D_d(N)le N^{1/d}Delta_d^{-1/d}, quad Nge 2,$$ where $Delta_d$ is the maximal sphere packing density in $R^d$. Further, we provide asymptotic estimates for the $f$-best-packing constants as $Ntoinfty$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
