M. Shapiro, V. Vinnikov, P. Yuditskii
We discuss a functional model for multi--diagonal selfadjoint operators with almost periodic coefficients that generalizes the well known model for finite band Jacobi matrices. It give us an opportunity to construct examples of almost periodic operators with different spectral properties. Main result deals with an exact condition for the uniqueness of the model of the given type.
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We construct multidimensional almost-periodic Schrödinger operators whose spectrum has zero lower box counting dimension. In particular, the spectrum in these cases is a generalized Cantor set of zero Lebesgue measure.
We consider some compact non-selfadjoint perturbations of fibered one-dimensional discrete Schrödinger operators. We show that the perturbed operator exhibits finite discrete spectrum under suitable\- regularity conditions.
Absence of flat bands for discrete periodic graph operators with generic potentials
该研究证明了离散连通周期图上的薛定谔型算子对于通用势不存在平带。
Time and band limiting operators are expressed as functions of the confluent Heun operator arising in the spheroidal wave equation. Explicit formulas are obtained when the bandwidth parameter is either small or large and results on the complete Fourier transform are recovered.
A model operator $H$ corresponding to a three-particle discrete Schrödinger operator on a lattice $\Z^3$ is studied. The essential spectrum is described via the spectrum of two Friedrichs models with parameters $h_α(p),$ $α=1,2,$ $p \in \T^3=(-π,π]^3.$ The following results are proven: 1) The operator $H$ has a finite number of eigenvalues lying below the bottom of the essential spectrum in any of the following cases: (i) both operators $h_α(0), α=1,2,$ have a zero eigenvalue; (ii) either $h_1(0)$ or $h_2(0)$ has a zero eigenvalue. 2) The operator $H$ has infinitely many eigenvalues lying belo…
We extend some basic results known for finite range operators to long range operators with off-diagonal decay. Namely, we prove an analogy of Sch'nol's theorem. We also establish the connection between the almost sure spectrum of long range random operators and the spectra of deterministic periodic operators.
We derive a new Chambers-type formula and prove sharper upper bounds on the measure of the spectrum of critical almost Mathieu operators with rational frequencies.
We present general principles for the preservation of a.c. spectrum under weak perturbations. The Schrodinger operators on the strip and on the Caley tree (Bethe lattice) are considered.
We consider a family of operators $-Δ+ t V$ with a slowly decaying and oscillating potential $V$. We prove that the absolutely continuous spectrum of this operator is essentially supported by $[0,\infty)$ for almost every $t$.
By approximation, I show that the spectrum of the Schrödinger operator with potential $V(n) = f(nρ\pmod 1)$ for f continuous and $ρ> 0$, $ρ\notin \N$ is an interval.
An example of Floquet operator with purely point spectrum and energy instability is presented. In the unperturbed energy eigenbasis its eigenfunctions are exponentially localized.
We show that there exists a dense set of frequencies with positive Hausdorff dimension for which the Hausdorff dimension of the spectrum of the critical almost Mathieu operator is positive.
We construct examples of potentials $V(x)$ satisfying $|V(x)| \leq \frac{h(x)}{1+x},$ where the function $h(x)$ is growing arbitrarily slowly, such that the corresponding Schrödinger operator has imbedded singular continuous spectrum. This solves one of the fifteen "twenty-first century" problems for Schrödinger operators posed by Barry Simon. The construction also provides the first example of a Schrödinger operator for which Möller wave operators exist but are not asymptotically complete due to the presence of singular continuous spectrum.
We give sufficient conditions for the presence of the absolutely continuous spectrum of a Schrödinger operator on a regular rooted tree without loops (also called regular Bethe lattice or Cayley tree).
This paper deals with the $L_p$-spectrum of Schrödinger operators on the hyperbolic plane. We establish Lieb-Thirring type inequalities for discrete eigenvalues and study their dependence on $p$. Some bounds on individual eigenvalues are derived as well.
We study the spectrum of random operators on a large class of trees. These trees have finitely many cone types and they can be constructed by a substitution rule. The random operators are perturbations of Laplace type operators either by random potentials or by random hopping terms, i.e., perturbations of the off-diagonal elements. We prove stability of arbitrary large parts of the absolutely continuous spectrum for sufficiently small but extensive disorder.
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.
We prove new criteria of stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators under slowly decaying perturbations. As applications, we show that the absolutely continuous spectrum of the free and periodic Schrödinger operators is preserved under perturbations by all potentials $V(x)$ satisfying $|V(x)| \leq C(1+x)^{-\frac{2}{3}-ε}.$ The main new technique includes an a.e.\ convergence theorem for a class of integral operators.
We prove that the essential spectrum of the operator obtained by linearization about a steady state of the Euler equations governing the motion of inviscid ideal fluid in dimension two is a vertical strip whose width is determined by the maximal Lyapunov exponent of the flow induced by the steady state.
We consider the one-dimensional Stark-Wannier type operators with potentials given by a smooth function with a logarithmic growth at infinity plus a periodic function with the Fourier coefficients of the form $(\ln |n|)^{-b}, 0<b<1/2$. We prove that in the case of rational electric field the spectrum of the corresponding operator is purely singular continuous.
We consider discrete periodic operator on $\mathbb Z^d$ with respect to lattices $Γ\subset\mathbb Z^d$ of full rank. We describe the class of lattices $Γ$ for which the operator may have a spectral gap for arbitrarily small potentials. We also show that, for a large class of lattices, the dimensions of the level sets of spectral band functions at the band edges do not exceed $d-2$.
In this paper we present a class of Anderson type operators with independent, non-stationary (non-decaying) random potentials supported on a subset of positive density in the odd-dimensional lattice and prove the existence of pure absolutely continuous spectrum in the middle of the band for small disorder.
We present a new oscillation criterion to determine whether the number of eigenvalues below the essential spectrum of a given Jacobi operator is finite or not. As an application we show that Kenser's criterion for Jacobi operators follows as a special case.
We consider a family of random Schrödinger operators on the discrete strip with decaying random $\ell^2$ matrix potential. We prove that the spectrum is almost surely pure absolutely continuous, apart from random, possibly embedded eigenvalues, which may accumulate at band edges.
We study the almost Mathieu operator at critical coupling. We prove that there exists a dense $G_δ$ set of frequencies for which the spectrum is of zero Hausdorff dimension.
We consider standard subordinacy theory for general Sturm--Liouville operators and give criteria when boundedness of solutions implies that no subordinate solutions exist. As applications, we prove a Weidmann-type result for general Sturm--Liouville operators and investigate the absolutely continuous spectrum of radially symmetric quantum trees.
We study multi-frequency quasiperiodic Schrödinger operators on $\mathbb{Z} $. We prove that for a large real analytic potential satisfying certain restrictions the spectrum consists of a single interval. The result is a consequence of a criterion for the spectrum to contain an interval at a given location that we establish non-perturbatively in the regime of positive Lyapunov exponent.
We study Schroedinger operators with periodic magnetic field in Euclidean 2-space, in the case of irrational magnetic flux. Positive measure Cantor spectrum is generically expected in the presence of an electric potential. We show that, even without electric potential, the spectrum has positive measure if the magnetic field is a perturbation of a constant one.
We consider a periodic system of domains coupled by small windows. In such domain we study the band spectrum of a Schroedinger operator subject to Neumann condition. We show that near each isolated eigenvalue of the similar operator but in the periodicity cell, there are several non-intersecting bands of the spectrum for the perturbed operator. We also discuss the position of the points at which the band functions attain the edges of each band.
We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon's list of Schrödinger operator problems for the twenty-first century.
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.
For a wide class of 2D periodic elliptic operators, we show that the global extrema of all spectral band functions are isolated.
We consider the quasi-periodic Schrödinger operator with the non-degenerate Gevrey potential for the Diophantine frequency. We prove that if the coupling number of the potential is large, then the spectrum is homogeneous.
We study a complex perturbation of a self-adjoint infinite band Schrodinger operator (defined in the form sense), and obtain the Lieb--Thirring type inequalities for the rate of convergence of the discrete spectrum of the perturbed operator to the joint essential spectrum of both operators.
The purpose of this paper is to prove that the spectrum of an isotropic Maxwell operator with electric permittivity and magnetic permeability that are periodic along certain directions and tending to a constant super-exponentially fast in the remaining directions is purely absolutely continuous. The basic technical tools is a new ``operatorial'' identity relating the Maxwell operator to a vector-valued Schrodinger operator. The analysis of the spectrum of that operator is then handled using ideas developed by the same authors in a previous paper.
If X is a contact Anosov vector field on a smooth compact manifold M and V is a smooth function on M, it is known that the differential operator A=-X+V has some discrete spectrum called Ruelle-Pollicott resonances in specific Sobolev spaces. We show that for |Im(z)| large the eigenvalues of A are restricted to vertical bands and in the gaps between the bands, the resolvent of A is bounded uniformly with respect to |Im(z)|. In each isolated band the density of eigenvalues is given by the Weyl law. In the first band, most of the eigenvalues concentrate of the vertical line Re(z)=< D >, the space…
We study the Schrödinger operator on $L_2(\mathbb R^3)$ with periodic variable metric, and periodic electric and magnetic fields. It is assumed that the operator is reflection symmetric and the (appropriately defined) flux of the magnetic field is rational. Under these assumptions it is shown that the spectrum of the operator is absolutely continuous. Previously known results on absolute continuity for periodic operators were obtained for the zero magnetic flux.
We give the first example of a smooth volume preserving mixing dynamical system such that the discrete Schrödinger operators on the line defined with a potential generated by this system and a Hölder sampling function, have almost surely a continuous spectrum.
The theory of random sets is demonstrated to prove useful for the theory of random operators. A random operator is here defined by requiring the graph to be a random set. It is proved that the spectrum and the set of eigenvalues of random operators are random sets. These results seem to be a novelty even in the case of random bounded operators. The main technical tools are given by the measurable selection theorem, the measurable projection theorem, and a characterisation of the spectrum by approximate eigenvalues of the operator and the adjoint operator. A discussion of some of the existing d…
We study the spectrum of random ergodic Schroedinger-type operators in the weak disorder regime. We give upper and lower bounds on how much the spectrum expands at its bottom for very general perturbations. The background operator is assumed to be a periodic elliptic differential operator on R^d, not necessarily of second order.
We prove Cwikel-Lieb-Rosenbljum and Lieb-Thirring type bounds on the discrete spectrum of a two-body pair operator and calculate spectral asymptotics for the eigenvalue moments and the local spectral density in the pseudo-relativistic limit.
We provide a description of the spectrum and essential spectra of invertible weighted composition operators acting in some algebras of smooth functions on the interval.
Spectrum of Weighted Composition Operators. Part IX. The spectrum and essential spectra of some weighted composition operators on uniform algebras
该研究在对应映射将Shilov边界映到自身的条件下，探究一致代数上加权复合算子的谱与上半Fredholm谱，得出解析一致代数下两类谱均具旋转不变性且谱为复平面连通子集的结论。
I prove that quasi-periodic Schrödinger operators in arbitrary dimension have some absolutely continuous spectrum.
Spectrum and Fine Spectrum of Band Matrices Generated by Oscillatory Sequences
本文研究由振荡序列生成的新型带状矩阵，用紧扰动技术分析其在ℓₚ（1<p<∞）空间的谱，得到多种谱细分结果及本质谱上无点谱的充分条件。
It is known that the spectrum of Schrödinger operators with sparse potentials consists of singular continuous spectrum. We give a sufficient condition so that the edge of the singular continuous spectrum is not an eigenvalue and construct examples with singular continuous spectrum which have no eigenvalues and which have a single negative eigenvalue.
I prove that the spectrum of a skew-shift Schrödinger operator contains larges interval in the semi-classical regime. In the semi-classical limit, these intervals approach the range of the potential.
We consider Schrödinger operators on periodic discrete graphs. It is known that the spectrum of these operators has band structure. We obtain a localization of spectral bands in terms of eigenvalues of Dirichlet and Neumann operators on a finite graph, which is constructed from the fundamental cell of the periodic graph. The proof is based on the Floquet decomposition of Schrödinger operators and the minimax principle.
We describe essential (in particular Fredholm and semi-Fredholm) spectra of operators on Banach lattices of the form $T=wU$, where $w$ is a central operator and $U$ is a disjointness preserving operator such that its spectrum $σ(U)$ is a subset of the unit circle.
On the spectrum of the differential operators of even order with periodic matrix coefficients
本文研究具有周期矩阵系数的偶数阶自伴微分算子的谱结构，给出了保证其谱中间隙数量有限的条件。
The Hecke operator associated with the Quaquaversal tiling, a highly anisotropic tiling introduced by Conway and Radin, is shown to have a real spectrum. Answering a question of Draco, Sadun, and Van Wieren. We also explicitly compute about three quarters of this operator's eigenvalues.
We prove $-Δ+V$ has purely discrete spectrum if $V\geq 0$ and, for all $M$, $|\{x\mid V(x)<M\}|<\infty$ and various extensions.
We characterize the spectrum and essential spectrum of "essentially linear fractional" composition operators acting on the Hardy space H-two of the open unit disc U. When the symbols of these composition operators have Denjoy-Wolff point on the unit circle, the spectrum and essential spectrum coincide. Our work permits us to describe the spectrum and essential spectrum of certain associated weighted composition operators on the Hardy space.
We establish equality between the essential spectrum of the Schroedinger operator with magnetic field in the exterior of a compact arbitrary dimensional domain and that of the operator defined in all the space, and discuss applications of this equality.
Global structure of the spectrum of periodic Non-hermitian Jacobi operators
本研究借助判别式及其驻点刻画周期非埃尔米特Jacobi算子谱的全局结构，同时给出该算子谱为实谱、单区间谱的充要条件。
For a class of zero order pseudodifferential operators we find the asymptotics of eigenvalues converging to a non-isolated tip of the essential spectrum.
Continuous movement of discrete spectrum of the Schrödinger operator $H(z)=-\frac{d^2} {dx^2}+V_0+z V_1$, with $\int_0^\infty {x |V_j(x)| dx} < \infty$, on the half-line is studied as $z$ moves along a continuous path in the complex plane. The analysis provides information regarding the members of the discrete spectrum of the non-selfadjoint operator that are evolved from the discrete spectrum of the corresponding selfadjoint operator.
We discuss the essential spectrum of essentially self-adjoint elliptic differential operators of first order and of Laplace type operators on Riemannian vector bundles over geometrically finite orbifolds.
On the spectrum of $2\times 2$ Dirac operator with degenerate boundary conditions
本文研究了具有退化边界条件和复值可和势的 $2\ imes 2$ Dirac 算子的谱问题，并找到了使该问题谱与无微扰算子谱一致的充分条件。
Absence of Cantor spectrum for a class of Schrödinger operators
结果表明，几乎Mathieu算子的特征向量的完全本地化需要该算子的Cantor谱的情况下。
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