We consider continuum random Schrödinger operators of the type $H_ω = -Δ+ V_0 + V_ω$ with a deterministic background potential $V_0$. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of $-Δ+V_0$. The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a one-dimensional surface (``random tube'') in arbitrary dimension.
We show that the spectral measure of discrete Schrödinger operators $ (Hu)(n)= u({n+1})+u({n-1})+V(n)u(n)$ does not have singular continuous component if the potential $V(n)=O(n^{-1})$.
We show that a generic quasi-periodic Schrödinger operator in $L^2(\mathbb{R})$ has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schrödinger operator with the resulting potential has empty absolutely continuous spectrum.
Fermi surfaces are basic objects in solid state physics and in the spectral theory of periodic operators. We define several measures connected to Fermi surfaces and study their measure theoretic properties. From this we get absence of singular continuous spectrum and of singular continuous components in the density of states for symmetric periodic elliptic differential operators acting on vector bundles. This includes Schroedinger operators with periodic magnetic field and rational flux, as well as the corresponding Pauli and Dirac-type operators.
A scattering zipper is a system obtained by concatenation of scattering events with equal even number of incoming and out going channels. The associated scattering zipper operator is the unitary equivalent of Jacobi matrices with matrix entries. For infinite identical events and random phases, Lyapunov exponents positivity is proved and yields to the absence of absolutely continuous spectrum.
Particle models with finitely many types of particles are considered, both on $\mathbb{Z}^d$ and on discrete point sets of finite local complexity. Such sets include many standard examples of aperiodic order such as model sets or certain substitution systems. The particle gas is defined by an interaction potential and a corresponding Gibbs measure. Under some reasonable conditions on the underlying point set and the potential, we show that the corresponding diffraction measure almost surely exists and consists of a pure point part and an absolutely continuous part with continuous density. In p…
The subject of the paper are Schrödinger operators on tree graphs which are radial having the branching number $b_n$ at all the vertices at the distance $t_n$ from the root. We consider a family of coupling conditions at the vertices characterized by $(b_n-1)^2+4$ real parameters. We prove that if the graph is sparse so that there is a subsequence of $\{t_{n+1}-t_n\}$ growing to infinity, in the absence of the potential the absolutely continuous spectrum is empty for a large subset of these vertex couplings, but on the the other hand, there are cases when the spectrum of such a Schrödinger ope…
This paper is a continuation of the paper \cite{W} by the third author, which studied quantum walks with special long-range perturbations of the coin operator. In this paper, we consider general long-range perturbations of the coin operator and prove the non-existence of a singular continuous spectrum and embedded eigenvalues. The proof relies on the construction of generalized eigenfunctions (Jost solutions) which was studied in the short-range case in \cite{MSSSSdis}.
We construct the one-dimensional analogous of von-Neumann Wigner potential to the relativistic Klein-Gordon operator, in which is defined taking asymptotic mathematical rules in order to obtain existence conditions of eigenvalues embedded in the continuous spectrum. Using our constructed potential, we provide an explicit and analytical example of the Klein-Gordon operator with positive eigenvalues embedded in the so called relativistic "continuum region". Even so in this not standard example, we present the region of the "continuum" where those eigenvalues cannot occur. Besides, the absence of…
We prove uniform absence of point spectrum for CMV operators corresponding to the period doubling subshift. We also prove almost sure absence of point spectrum for CMV operators corresponding to a class of Sturmian subshifts. Lastly, we prove almost sure absence of point spectrum for CMV operators corresponding to some subshifts generated by a coding of a rotation.
Quasiperiodic tilings are often considered as structure models of quasicrystals. In this context, it is important to study the nature of the diffraction measures for tilings. In this article, we investigate the diffraction measures for S-adic tilings in R^d, which are constructed from a family of geometric substitution rules. In particular, we firstly give a sufficient condition for the absolutely continuous component of the diffraction measure for an S-adic tiling to be zero. Next, we prove this sufficient condition for "almost all" binary block-substitution cases and thus prove the absence o…
The pair correlations of primitive inflation rules are analysed via their exact renormalisation relations. We introduce the inflation displacement algebra that is generated by the Fourier matrix of the inflation and deduce various consequences of its structure. Moreover, we derive a sufficient criterion for the absence of absolutely continuous diffraction components, as well as a necessary criterion for its presence. This is achieved via estimates for the Lyapunov exponents of the Fourier matrix cocycle of the inflation rule. We also discuss some consequences for the spectral measures of such …
The absolutely continuous spectrum of an ergodic family of one-dimensional Schrödinger operators is completely determined by the Lyapunov exponent as shown by Ishii, Kotani and Pastur. Moreover, the part of the theory developed by Kotani gives powerful tools for proving the absence of absolutely continuous spectrum, the presence of absolutely continuous spectrum, and even the presence of purely absolutely continuous spectrum. We review these results and their recent applications to a number of problems: the absence of absolutely continuous spectrum for rough potentials, the absence of absolute…
We consider Schödinger operators on the half-line, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if $Δ+ V$ has no spectrum outside of the interval $[-2,2]$, then it has purely absolutely continuous spectrum. In the continuum case we show that if both $-Δ+ V$ and $-Δ- V$ have no spectrum outside $[0,\infty)$, then both operators are purely absolutely continuous. These results extend to operators with finitely many bound states.
A new continuous regularized Gauss-Newton-type method with simultaneous updates of the operator $(F^{\pr*}(x(t))F'(x(t))+\ep(t) I)^{-1}$ for solving nonlinear ill-posed equations in a Hilbert space is proposed. A convergence theorem is proved. An attractive and novel feature of the proposed method is the absence of the assumptions about the location of the spectrum of the operator $F'(x)$. The absence of such assumptions is made possible by a source-type condition.
The problem of absence of eigenvalues imbedded into the continuous spectrum is considered for a Schrödinger operator with a periodic potential perturbed by a sufficiently fast decaying ``impurity'' potential. Results of this type have previously been known for the one-dimensional case only. Absence of embedded eigenvalues is shown in dimensions two and three if the corresponding Fermi surface is irreducible modulo natural symmetries. It is conjectured that all periodic potentials satisfy this condition. Separable periodic potentials satisfy it, and hence in dimensions two and three Schrödinger…
Wrocław University of Science and Technology（弗罗茨瓦夫科技大学）；Laboratoire de Mathématiques Pure et Appliquées Centre Universitaire de la Mi-Voix（纯粹与应用数学实验室中部大学中心）
We consider continuous-spin models on the $d$-dimensional hypercubic lattice with the spins $σ_x$ \emph{a priori} uniformly distributed over the unit sphere in $\R^n$ (with $n\ge2$) and the interaction energy having two parts: a short-range part, represented by a potential $Φ$, and a long-range antiferromagnetic part $λ|x-y|^{-s}σ_x\cdotσ_y$ for some exponent $s>d$ and $λ\ge0$. We assume that $Φ$ is twice continuously differentiable, finite range and invariant under rigid rotations of all spins. For $d\ge1$, $s\in(d,d+2]$ and any $λ>0$, we then show that the expectation of each $σ_x$ vanishes …
We study a continuous matrix-valued Anderson-type model. Both leading Lyapunov exponents of this model are proved to be positive and distinct for all ernergies in $(2,+\infty)$ except those in a discrete set, which leads to absence of absolutely continuous spectrum in $(2,+\infty)$. This result is an improvement of a previous result with Stolz. The methods, based upon a result by Breuillard and Gelander on dense subgroups in semisimple Lie groups, and a criterion by Goldsheid and Margulis, allow for singular Bernoulli distributions.
The water wave problem is considered for a class of semi-infinite domains each having its shore shaped as a cliffed cape. In particular, the free surface of a water domain is supposed to be an infinite sector whose vertex angle is greater than $π$, whereas the water layer lying under the free surface is of constant depth. Under these assumptions, it is shown that there are no trapped mode solutions of the problem for all values of a non-dimensional spectral parameter; in other words, no point eigenvalues are embedded in the problem's continuous spectrum.
We present a result of absence of absolutely continuous spectrum in an interval of $\R$, for a matrix-valued random Schrödinger operator, acting on $L^2(\R)\otimes \R^N$ for an arbitrary $N\geq 1$, and whose interaction potential is generic in the real symmetric matrices. For this purpose, we prove the existence of an interval of energies on which we have separability and positivity of the $N$ non-negative Lyapunov exponents of the operator. The method, based upon the formalism of Fürstenberg and a result of Lie group theory due to Breuillard and Gelander, allows an explicit contruction of the…
We show the absence of continuous symmetry breaking in 2D lattice systems without any smoothness assumptions on the interaction. We treat certain cases of interactions with integrable singularities. We also present cases of singular interactions with continuous symmetry, when the symmetry is broken in the thermodynamic limit.
We prove a criterion for absence of eigenvalues for one-dimensional Schrödinger operators. This criterion can be regarded as an $L^1$-version of Gordon's theorem and it has a broader range of application. Absence of eigenvalues is then established for quasiperiodic potentials generated by Liouville frequencies and various types of functions such as step functions, Hölder continuous functions and functions with power-type singularities. The proof is based on Gronwall-type a priori estimates for solutions of Schrödinger equations.
We consider CMV matrices with dynamically defined Verblunsky coefficients. These coefficients are obtained by continuous sampling along the orbits of an ergodic transformation. We investigate whether certain spectral phenomena are generic in the sense that for a fixed base transformation, the set of continuous sampling functions for which the spectral phenomenon occurs is residual. Among the phenomena we discuss are the absence of absolutely continuous spectrum and the vanishing of the Lebesgue measure of the spectrum.
In this note, we study a continuous matrix-valued Anderson-type model. Both leading Lyapounov exponents of this model are proved to be positive and distincts for all energies in $(2,+\infty)$ except those in a discrete set, which leads to absence of absolutely continuous spectrum in $(2,+\infty)$. The methods, using group theory results by Breuillard and Gelander, allow for singular Bernoulli distributions.
We study the multi-particle Anderson model in the continuum and show that under some mild assumptions on the random external potential and the inter-particle interaction, for any finite number of particles, the multi-particle lower edges of the spectrum are almost surely constant in absence of ergodicity. We stress that this result is not quite obvious and has to be handled carefully. In addition, we prove the spectral exponential and the strong dynamical localization of the continuous multi-particle Anderson model at low energy. The proof based on the multi-particle multi-scale analysis bound…
