Sturm–Liouville operators in the half axis with shifted potentials

We consider Sturm–Liouville operators in the half axis generated by shifts of the potential and prove that Lebesgue measure is equivalent to a measure defined as an average of spectral measures which correspond to these operators. This is then used to obtain results on stability of spectral types under change of parameters such as boundary conditions, local perturbations, and shifts. In particular if for a set of shifts of positive measure the corresponding operators have α-singular, singular continuous and (or) point spectrum in a fixed interval, then this set of shifts has to be unbounded. Moreover, there are large sets of boundary conditions and local perturbations for which the corresponding operators enjoy the same property.     

On the spectral theory of trees with finite cone type

We study basic spectral features of graph Laplacians associated with a class of rooted trees which contains all regular trees. Trees in this class can be generated by substitution processes. Their spectra are shown to be purely absolutely continuous and to consist of finitely many bands. The main result gives stability of the absolutely continuous spectrum under sufficiently small radially label symmetric perturbations for non-regular trees in this class. In sharp contrast, the absolutely continuous spectrum can be completely destroyed by arbitrary small radially label symmetric perturbations for regular trees in this class.     

Constancy of spectra of equivariant (non-selfadjoint) operators over minimal dynamical systems

We show that for a large class of equivariant continuous, possibly non-selfadjoint, operators over minimal dynamical systems the spectrum as a set is in fact independent of the random parameter. Furthermore, the spectrum agrees with the essential spectrum. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Large deviations for a symmetric branching random walk on a multidimensional lattice

An important role in the theory of branching random walks is played by the problem of the spectrum of a bounded symmetric operator, the generator of a random walk on a multidimensional integer lattice, with a one-point potential. We consider operators with potentials of a more general form that take nonzero values on a finite set of points of the integer lattice. The resolvent analysis of such operators has allowed us to study branching random walks with large deviations. We prove limit theorems on the asymptotic behavior of the Green function of transition probabilities. Special attention is paid to the case when the spectrum of the evolution operator of the mean numbers of particles contains a single eigenvalue. The results obtained extend the earlier studies in this field in such directions as the concept of a reaction front and the structure of a population inside a front and near its boundary.     

Perturbation of spectra and spectral subspaces

We consider the problem of variation of spectral subspaces for linear self-adjoint operators with emphasis on the case of off-diagonal perturbations. We prove a number of new optimal results on the shift of the spectrum and obtain (sharp) estimates on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators, respectively.

     

Weak Disorder Localization and Lifshitz Tails: Continuous Hamiltonians

This paper is devoted to the study of band edge localization for continuous random Schrödinger operators with weak random perturbations. We prove that, in the weak disorder regime, l \lambda small, the spectrum in a neighborhood of size C ·l C \cdot \lambda of a non-degenerate simple band edge is exponentially and dynamically localized. Upper bounds on the localization length in these energy regions are also obtained. Our results rely on the analysis of Lifshitz tails when the disorder is small; the single site potential need not be of fixed sign.     

Eigenvalue Asymptotics for Locally Perturbed Second-Order Differential Operators

We consider the appearance of discrete spectrum in spectralgaps of magnetic Schrödinger operators with electric backgroundfield under strong, localised perturbations. We show that forcompactly supported perturbations the asymptotics of the countingfunction of the occurring eigenvalues in the limit of a strongperturbation does not depend on the magnetic field nor on thebackground field.     

Proof of dynamical localization for perturbations of discrete 1D Schrödinger operators with uniform electric fields

Mathematische Zeitschrift - We present a proof of discrete spectrum and dynamical localization for small perturbations of discrete one-dimensional Schrödinger operators with uniform electric...     

Stability of spectral types for Jacobi matrices under decaying random perturbations

We study stability of spectral types for semi-infinite self-adjoint tridiagonal matrices under random decaying perturbations. We show that absolutely continuous spectrum associated with bounded eigenfunctions is stable under Hilbert-Schmidt random perturbations. We also obtain some results for singular spectral types.     

Perturbed kernel approximation on homogeneous manifolds

Current methods for interpolation and approximation within a native space rely heavily on the strict positive-definiteness of the underlying kernels. If the domains of approximation are the unit spheres in euclidean spaces, then zonal kernels (kernels that are invariant under the orthogonal group action) are strongly favored. In the implementation of these methods to handle real world problems, however, some or all of the symmetries and positive-definiteness may be lost in digitalization due to small random errors that occur unpredictably during various stages of the execution. Perturbation analysis is therefore needed to address the stability problem encountered. In this paper we study two kinds of perturbations of positive-definite kernels: small random perturbations and perturbations by Dunkl's intertwining operators [C. Dunkl, Y. Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and Its Applications, vol. 81, Cambridge University Press, Cambridge, 2001]. We show that with some reasonable assumptions, a small random perturbation of a strictly positive-definite kernel can still provide vehicles for interpolation and enjoy the same error estimates. We examine the actions of the Dunkl intertwining operators on zonal (strictly) positive-definite kernels on spheres. We show that the resulted kernels are (strictly) positive-definite on spheres of lower dimensions.     

The Integrated Density of States for Some Random Operators with Nonsign Definite Potentials

We study the integrated density of states of random Anderson-type additive and multiplicative perturbations of deterministic background operators for which the single-site potential does not have a fixed sign. Our main result states that, under a suitable assumption on the regularity of the random variables, the integrated density of states of such random operators is locally Hölder continuous at energies below the bottom of the essential spectrum of the background operator for any nonzero disorder, and at energies in the unperturbed spectral gaps, provided the randomness is sufficiently small. The result is based on a proof of a Wegner estimate with the correct volume dependence. The proof relies upon the L^p-theory of the spectral shift function for p?1 (Comm. Math. Phys.218 (2001), 113-130), and the vector field methods of Klopp (Comm. Math. Phys.167 (1995), 553-569). We discuss the application of this result to Schrödinger operators with random magnetic fields and to band-edge localization.     

Nonself-adjoint operators with almost Hermitian spectrum: Matrix model. I

Nonself-adjoint, nondissipative perturbations of bounded self-adjoint operators with real purely singular spectrum are considered. Using a functional model of a nonself-adjoint operator as a principal tool, spectral properties of such operators are investigated. In particular, in the case of rank two perturbations the pure point spectral component is completely characterized in terms of matrix elements of the operators’ characteristic function.     

Property （ω） and topological uniform descent

We give the necessary and sufficient condition for a bounded linear operator with property （ω） by means of the induced spectrum of topological uniform descent, and investigate the permanence of property （ω） under some commuting perturbations by power finite rank operators. In addition, the theory is exemplified in the case of algebraically paranormal operators.     

Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations

In this article we study global-in-time Strichartz estimates for the Schrödinger evolution corresponding to long-range perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article [D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math. 130 (2008) 571-634] of the third author, where it is proved that local smoothing estimates imply Strichartz estimates. By [D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math. 130 (2008) 571-634] the local smoothing estimates are known to hold for small perturbations of the Laplacian. Here we consider the case of large perturbations in three increasingly favorable scenarios: (i) without non-trapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term, (ii) with non-trapping assumptions we prove global estimates modulo a lower order spatially localized error term, and (iii) for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum.     

On the preservation of absolutely continuous spectrum for Schrödinger operators

We present general principles for the preservation of a.c. spectrum under weak perturbations. The Schrödinger operators on the strip and on the Caley tree (Bethe lattice) are considered.     

Resonant Delocalization on the Bethe Strip

Recently, Aizenman and Warzel discovered a mechanism for the appearance of absolutely continuous spectrum for random Schrödinger operators on the Bethe lattice through rare resonances (resonant delocalization). We extend their analysis to operators with matrix-valued random potentials drawn from ensembles such as the Gaussian Orthogonal Ensemble. These operators can be viewed as random operators on the Bethe strip, a graph (lattice) with loops.     

A characterization of some subsets of Schechter's essential spectrum and application to singular transport equation

The purpose of this paper is to provide a detailed treatment of some subsets of Schechter's essential spectrum of closed, densely defined linear operators subjected to additive perturbations. Our results are used to describe the essential approximate point spectrum and the essential defect spectrum of singular neutron transport operators in bounded geometries.     

Isometries, Fock spaces, and spectral analysis of Schrödinger operators on trees

We construct conjugate operators for the real part of a completely non-unitary isometry and we give applications to the spectral and scattering theory of a class of operators on (complete) Fock spaces, natural generalizations of the Schrödinger operators on trees. We consider C^*-algebras generated by such Hamiltonians with certain types of anisotropy at infinity, we compute their quotient with respect to the ideal of compact operators, and give formulas for the essential spectrum of these Hamiltonians.     

Stability of essential spectra of singular Sturm‐Liouville differential operators under perturbations small at infinity

This paper is concerned with the stability of essential spectra of singular Sturm‐Liouville differential operators with complex‐valued coefficients. It is proved that the essential spectrum of the corresponding minimal operator is preserved by perturbations small at infinity with respect to the unperturbed operator. Based on it, 1‐dimensional Schrödinger operators under local dilative perturbations are studied.     

Perturbation of semi-Browder operators and stability of Browder's essential defect and approximate point spectrum

In the present paper we investigate the stability of closed densely defined semi-Browder operators under operator perturbations that belong to a perturbation class related to compact operators. Furthermore, we apply the obtained results to give a characterization and to study the stability of Browder's essential approximate point spectrum and Browder's essential defect spectrum.