Absolutely continuous spectrum of fourth order difference equations

We have studied the absolutely continuous spectrum of a selfadjoint subspace extension generated by a three‐term fourth order difference equation with bounded coefficients using subspace theory. In particular, we have shown that the absolutely continuous spectrum exists outside a certain bounded interval. In addition, we have computed the spectral multiplicity as well as the location of absolutely continuous spectrum of selfadjoint subspace extension under certain asymptotic conditions.     

Spectral theory of difference operators with almost constant coefficients II

We show that the operators whose coefficients are approximately constant in a general sense have an absolutely continuous spectrum which is equal to that of the corresponding constant coefficient operator. For such operators, the absolutely continuous spectrum can be read off from the associated characteristic polynomial. This generalizes the classical results on second-order operators and extends those of higher order differential operators to the difference setting. Our approach relies on an analysis of the associated difference equation with the help of uniform asymptotic summation techniques.     

Absolutely continuous spectrum of perturbed Stark operators

We prove new results on the stability of the absolutely continuous spectrum for perturbed Stark operators with decaying or satisfying certain smoothness assumption perturbation. We show that the absolutely continuous spectrum of the Stark operator is stable if the perturbing potential decays at the rate or if it is continuously differentiable with derivative from the Hölder space with any

     

Spectral theory of difference operators with almost constant coefficients

The spectrum of higher even order difference operators with almost constant coefficients is determined. With appropriate smoothness and decay conditions on the coefficients, we show that singular continuous spectrum is absent and that the absolutely continuous spectrum agrees with that of the constant coefficient limiting operator. For such operators, the absolutely continuous spectrum is determined uniquely by the range of the characteristic polynomial. This result extends a similar result for even order differential operators. The methods of proof are closely related likewise. Finally, some results on fourth order operators with unbounded coefficients are shown.     

Absolutely continuous spectrum for random Schrödinger operators on the Bethe strip

The Bethe strip of width m is the cartesian product $\mathbb {B}\times \lbrace 1,\ldots ,m\rbrace$, where $\mathbb {B}$ is the Bethe lattice (Cayley tree). We prove that Anderson models on the Bethe strip have “extended states” for small disorder. More precisely, we consider Anderson‐like Hamiltonians $H_\lambda =\frac{1}{2} \Delta \otimes 1 + 1 \otimes A\,+\,\lambda \mathcal {V}$ on a Bethe strip with connectivity K ≥ 2, where A is an m × m symmetric matrix, $\mathcal {V}$ is a random matrix potential, and λ is the disorder parameter. Given any closed interval $I\subset \big (\!-\!\sqrt{K}+a_{{\rm max}},\sqrt{K}+a_{\rm {min}}\big )$, where a_min and a_max are the smallest and largest eigenvalues of the matrix A, we prove that for λ small the random Schrödinger operator H_λ has purely absolutely continuous spectrum in I with probability one and its integrated density of states is continuously differentiable on the interval I.     

Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results

The absolutely continuous spectrum of one-dimensional Schrödinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectra of free and periodic Schrödinger operators are preserved under all perturbations satisfying This result is optimal in the power scale. Slightly more general classes of perturbing potentials are also treated. A general criterion for stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators is established. In all cases analyzed, the main term of the asymptotic behavior of the generalized eigenfunctions is shown to have WKB form for almost all energies. The proofs rely on maximal function and norm estimates, and on almost everywhere convergence results for certain multilinear integral operators.

     

Absolutely Continuous Spectrum for Multi-type Galton Watson Trees

We consider multi-type Galton Watson trees that are close to a tree of finite cone type in distribution. Moreover, we impose that each vertex has at least one forward neighbor. Then, we show that the spectrum of the Laplace operator exhibits almost surely a purely absolutely continuous component which is included in the absolutely continuous spectrum of the tree of finite cone type.     

Scattering Matrix for Magnetic Potentials with Coulomb Decay at Infinity

We consider the Schrödinger operator H in the space $ L_{2}(\mathbb{R}^{d})$ with a magnetic potential A(x) decaying as $ \vert x\vert^{-1} $ at infinity and satisfying the transversal gauge condition <A(x), x > = 0. Our goal is to study properties of the scattering matrix S() associated to the operator H. In particular, we find the essential spectrum _ess of S() in terms of the behaviour of A(x) at infinity. It turns out that _ess(S()) is normally a rich subset of the unit circle $\mathbb{T}$ or even coincides with $\mathbb{T}$. We find also the diagonal singularity of the scattering amplitude (of the kernel of S() regarded as an integral operator). In general, the singular part S₀ of the scattering matrix is a sum of a multiplication operator and of a singular integral operator. However, if the magnetic field decreases faster than $ \vert x\vert^{-1} $ for d 3 (and the total magnetic flux is an integer times 2 for dd = 2), then this singular integral operator disappears. In this case the scattering amplitude has only a weak singularity (the diagonal Dirac function is neglected) in the forward direction and hence scattering is essentially of short-range nature. Moreover, we show that, under such assumptions, the absolutely continuous parts of the operators S() and S₀ are unitarily equivalent. An important point of our approach is that we consider S() as a pseudodifferential operator on the unit sphere and find an explicit expression of its principal symbol in terms of A(x). Another ingredient is an extensive use (for d 3) of a special gauge adapted to a magnetic potential A(x).     

Substitution-based structures with absolutely continuous spectrum

By generalising Rudin’s construction of an aperiodic sequence, we derive new substitution-based structures which have a purely absolutely continuous diffraction measure and a mixed dynamical spectrum, with absolutely continuous and pure point parts. We discuss several examples, including a construction based on Fourier matrices which yields constant-length substitutions for any length.     

Linear invariant measures for recurrent linear systems

We consider a self-adjoint operator defined by a bidimensional linear system. We extend the Ishii-Pastur-Kotani theory that allows us to identify the absolutely continuous spectrum. From here we deduce that for almost everyE with null Lyapunov exponent the real projective flow admits absolutely continuous invariant measures with square integrable density function.     

ʧЧ��������ϵͳ���������Ԫ������

??Coherent systems are very important in reliability,survival analysis and other life sciences. In this paper, we consider the number of working components in an $(n-k+1)$-out-of-$n$ system, given that at least $(n-m+1)$ components are working at time $t$, and the system has failed at time $t$. In this condition, we compute the probability that there are exactly $i$ working components. First the reliability and several stochastic properties are obtained. Furthermore, we extend the results to general coherent systems with absolutely continuous and exchangeable components.     

One-dimensional Schrödinger operators with random or deterministic potentials: New spectral types

We prove sufficient conditions involving only potential asymptotic near one of the infinities in order to have purely absolutely continuous components in the spectrum. These deterministic results are then applied to random cases and exhibit classes of models for which, with probability one, one component of the spectrum is purely absolutely continuous and the rest is dense pure point with exponentially decaying eigenfunctions.     

单调关联系统的条件失效元件数量

Coherent systems are very important in reliability,survival analysis and other life sciences.In this paper,we consider the number of failed components in an(n-k+1)-out-of-n system,given that at least m(m相似文献   

Absolutely continuous spectrum of Stark operators

We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely continuous spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is |q(x)|≤C(1+|x|)^?1/4?ε; in the smoothness direction, a sufficient condition in Hölder classes isq∈C^1/2+ε(R). On the other hand, we show that there exist potentials which both satisfy |q(x)|≤C(1+|x|)^?1/4 and belong toC^1/2(R) for which the spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.     

Essential and continuous spectrum of symmetric difference equations

Essential and continuous spectrum of symmetric difference equations have been investigated. It has been shown that the deficiency indices and the existence of these components of the spectrum are determined by the growth conditions of the coefficients. In particular, the deficiency indices are superimposition of those clusters determined by the coefficient growth. Finally, we have proved the neccessary and sufficient conditions for the existence of essential spectrum of selfadjoint subspace extensions using subspace theory and asymtotic summation.     

Spectral theory for slowly oscillating potentials I. Jacobi matrices

For Jacobi matrices on the discrete half line with slowly oscillating potentials the absolutely continuous and singular spectrum is located. The results, which can be extended to perturbed periodic potentials, show that separated regions of purely absolutely continuous resp. purely singular spectrum appear. The main tools in the proof of absolute continuity are the method of subordinacy and an abstract result on the iterated diagonalization of products of 2×2-matrices, which is applied to transfer matrices. The singular spectrum is found by using a result of Simon and Spancer.     

Refinement equations with nonnegative coefficients

In this paper we analyze solutions of the n-scale functional equation Ф(x) = Σ_k∈ℤ ^Pk Ф(nx−k), where n≥2 is an integer, the coefficients {P_k} are nonnegative and Σpk = 1. We construct a sharp criterion for the existence of absolutely continuous solutions of bounded variation. This criterion implies several results concerning the problem of integrable solutions of n-scale refinement equations and the problem of absolutely continuity of distribution function of one random series. Further we obtain a complete classification of refinement equations with positive coefficients (in the case of finitely many terms) with respect to the existence of continuous or integrable compactly supported solutions.     

Natural boundaries and spectral theory

We present and exploit an analogy between lack of absolutely continuous spectrum for Schrödinger operators and natural boundaries for power series. Among our new results are generalizations of Hecke's example and natural boundary examples for random power series where independence is not assumed.     

The Green’s Function in the Problem of Charge Dynamics on A One-Dimensional Lattice with An Impurity Center

In the “tight-binding” approximation (the Hückel model), we consider the evolution of the charge wave function on a semi-infinite one-dimensional lattice with an additional energy U at a single impurity site. In the case of the continuous spectrum (for |U| < 1) where there is no localized state, we construct the Green’s function using the expansion in terms of eigenfunctions of the continuous spectrum and obtain an expression for the time Green’s function in the form of a power series in U. It unexpectedly turns out that this series converges absolutely even in the case where the localized state is added to the continuous spectrum. We can therefore say that the Green’s function constructed using the states of the continuous spectrum also contains an implicit contribution from the localized state.