On the Absolutely Continuous Spectrum of Dirac Operator

Abstract

We prove that the massless Dirac operator in ?³ with long-range potential has an a.c. spectrum which fills the whole real line. The Dirac operators with matrix-valued potentials are considered as well.     

Absolutely Continuous Spectrum for the Anderson Model on Some Tree-like Graphs

We prove persistence of absolutely continuous spectrum for the Anderson model on a general class of tree-like graphs.     

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.     

Absolutely Continuous Spectrum of a Polyharmonic Operator with a Limit Periodic Potential in Dimension Two

We consider a polyharmonic operator H = (?Δ) ^l  + V(x) in dimension two with l ≥ 6, l being an integer, and a limit-periodic potential V(x). We prove that the spectrum contains a semiaxis of absolutely continuous spectrum.     

Absence of Absolutely Continuous Spectrum for the Kirchhoff Laplacian on Radial Trees

In this paper, we prove that the existence of absolutely continuous spectrum of the Kirchhoff Laplacian on a radial metric tree graph together with a finite complexity of the geometry of the tree implies that the tree is in fact eventually periodic. This complements the results by Breuer and Frank in (Rev Math Phys 21(7):929–945, 2009) in the discrete case as well as for sparse trees in the metric case.     

On the Spectral Gap of a Quantum Graph

We consider the problem of finding universal bounds of “isoperimetric” or “isodiametric” type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature.     

Schur Complement and Continuous Spectrum in a Kinetic Plasma Model

Doklady Mathematics - The goal of this paper is spectral analysis of an evolutionary semigroup generator describing the dynamics of a rarefied two-component plasma subjected to a self-consistent...     

Absolutely Continuous Embeddings of Rearrangement-Invariant Spaces

Compactness of embeddings between rearrangement invariant spaces is closely related to absolute continuity of these embeddings. We study absolutely continuous embeddings between rearrangement invariant spaces. In particular it is shown that an absolutely continuous embedding is never optimal. We give sufficient (and under additional hypotheses necessary) conditions for absolute continuity of these embeddings. We also provide quantitative estimates of absolutely continuous embeddings.     

Embedded Spectrum on a Metric Graph (an Observation)

We suggest an elementary example of a simplest starlike graph that shows that the analytical disjunction may occur not on only at isolated points of the spectral parameter, but also on whole intervals. Bibliography: 1 title.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 215–220     

On the Number of All Absolutely Continuous,Ergodic Measures of MARKOV Type Transformations Defined on an Interval

An effective procedure is given to get all absolutely continuous, ergodic measures of MARKOV type transformations defined on an interval.     

On the Discrete Spectrum of Generalized Quantum Tubes

The spectrum (of the Dirichlet Laplacian) of non-compact, non-complete Riemannian manifolds is much less understood than their compact counterparts. In particular it is often not even known whether such a manifold has any discrete spectra. In this article, we will prove that a certain type of non-compact, non-complete manifold called the quantum tube has non-empty discrete spectrum. The quantum tube is a tubular neighborhood built about an immersed complete manifold in Euclidean space. The terminology of “quantum” implies that the geometry of the underlying complete manifold can induce discrete spectra – hence quantization. We will show how the Weyl tube invariants appear in determining the existence of discrete spectra. This is an extension and generalization, on the geometric side, of the previous work of the author on the “quantum layer.”     

On the Spectrum of the Resonant Quantum Kicked Rotor

It is proven that none of the bands in the quasi-energy spectrum of the Quantum Kicked Rotor is flat at any primitive resonance of any order. Perturbative estimates of bandwidths at small kick strength are established for the case of primitive resonances of prime order. Different bands scale with different powers of the kicking strength, due to degeneracies in the spectrum of the free rotor.     

绝对连续测度的Loeb代数

It is proved that if μ and v be finite measures on a measurable space (X,S) and v is absolutely continuous with respect to v, then it is holds that L(^*S,^*v) C L(^*S,^*v), while L(^*S,^*v) and L(^*S,^*v) are the Loeb algebras with respect to measure spaces (X,S,μ) and(X,S,μ).     

On the Spectrum of 1D Quantum Ising Quasicrystal

We consider one-dimensional quantum Ising spin-1/2 chains with two-valued nearest neighbor couplings arranged in a quasi-periodic sequence, with uniform, transverse magnetic field. By employing the Jordan–Wigner transformation of the spin operators to spinless fermions, the energy spectrum can be computed exactly on a finite lattice. By employing the transfer matrix technique and investigating the dynamics of the corresponding trace map, we show that in the thermodynamic limit the energy spectrum is a Cantor set of zero Lebesgue measure. Moreover, we show that local Hausdorff dimension is continuous and non-constant over the spectrum. This forms a rigorous counterpart of numerous numerical studies.     

C－余弦算子函数的谱

Let A be the generator of a C-cosine operator functions C(t) on a Banach space X .The spectrum characterization of (unbounded) cosine operator function C(t)C^-1 on R(C²) is derived. In particular， we characterize the spectrum of strongly continuous cosine operator on a Banach space.     

On the Maximum Matching Graph of a Graph

1IntroductionMatchingtheory,aswellastheassignmentprobleminlinearprogramming,hasawiderangeofapplicationinthetheoryandpracticeofoperationsresearch.Bysomepracticalmotivations,e.g.,forfindingalloptimalsolutions,peoplewanttoknowthestructurepropertiesofallmaximummatchingsofagraphG.InthecasethatGhasperfectmatchings,extensiveworkhasbeendoneontheso-calledperfectmatChinggrape(or1-factorgraph),inwhichtwoperfectmatchingsMIandMZaresaidtobeadjacentifMI~MZ@E(C)whereCisanMI-alternatingcycleofG.Therewer…     

一类可靠性模型的主算子的谱

研究一个可靠机器,一个具有无穷容量的缓冲库和一个不可靠机器构成的系统的主算子的谱.当机器1的加工率λ,机器2的加工率μ,部件从机器1成功地传送到机器2的概率η₁,机器2成功处理完部件的概率η₂满足一定的条件时,证明-μ不是该系统的主算子的特征值.     

On the Peripheral Spectrum of Order Continuous,Positive Operators

Let E be an order complete Banach function lattice and T a positive, eventually compact, order continuous operator on E. We study necessary conditions under which the peripheral spectrum of T is fully cyclic in terms of certain bands of the underlying Banach function lattice E. A set of sufficient conditions is also given. Examples are presented to demonstrate our methods.