Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the spatial extent of localized eigenfunctions for random Schrodinger operators
113
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The present paper is devoted to new, improved bounds for the eigenfunctions of random operators in the localized regime. We prove that, in the localized regime with good probability, each eigenfunction is exponentially decaying outside a ball of a certain radius, which we call the localization onset length. For $ell>0$ large, we count the number of eigenfunctions having onset length larger than $ell$ and find it to be smaller than $exp(-Cell)$ times the total number of eigenfunctions in the system. Thus, most eigenfunctions localize on finite size balls independent of the system size.
rate research
Read More
This paper studies the delocalized regime of an ultrametric random operator whose independent entries have variances decaying in a suitable hierarchical metric on $mathbb{N}$. When the decay-rate of the off-diagonal variances is sufficiently slow, we prove that the spectral measures are uniformly $theta$-H{o}lder continuous for all $theta in (0,1)$. In finite volumes, we prove that the corresponding ultrametric random matrices have completely extended eigenfunctions and that the local eigenvalue statistics converge in the Wigner-Dyson-Mehta universality class.
We demonstrate how the Moutard transformation of two-dimensional Schrodinger operators acts on the Faddeev eigenfunctions on the zero energy level and present some explicitly computed examples of such eigenfunctions for smooth fast decaying potentials of operators with non-trivial kernel and for deformed potentials which correspond to blowing up solutions of the Novikov-Veselov equation.
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.
We use trace class scattering theory to exclude the possibility of absolutely continuous spectrum in a large class of self-adjoint operators with an underlying hierarchical structure and provide applications to certain random hierarchical operators and matrices. We proceed to contrast the localizing effect of the hierarchical structure in the deterministic setting with previous results and conjectures in the random setting. Furthermore, we survey stronger localization statements truly exploiting the disorder for the hierarchical Anderson model and report recent results concerning the spectral statistics of the ultrametric random matrix ensemble.
We consider Schrodinger operators on [0,infty) with compactly supported, possibly complex-valued potentials in L^1([0,infty)). It is known (at least in the case of a real-valued potential) that the location of eigenvalues and resonances determines the potential uniquely. From the physical point of view one expects that large resonances are increasingly insignificant for the reconstruction of the potential from the data. In this paper we prove the validity of this statement, i.e., we show conditional stability for finite data. As a by-product we also obtain a uniqueness result for the inverse resonance problem for complex-valued potentials.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
