Related papers: Borg type uniqueness Theorems for periodic Jacobi …
Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals and in more general studies of structures exhibiting aperiodic order. The spectra of these self-adjoint operators can be quite exotic, such as Cantor sets, and…
We discuss the notion of Jacobi forms of degree one with matrix index, we state dimension formulas, give explicit examples, and indicate how closely their theory is connected to the theory of invariants of Weil representations associated to…
An alternative proof of the duality of generalized Lie bialgebroid is given and proved a canonical Jacobi structure can be defined on the base of it. We also introduce the notion of morphism between generalized Lie bialgebroids and proved…
We develop relative oscillation theory for Jacobi matrices which, rather than counting the number of eigenvalues of one single matrix, counts the difference between the number of eigenvalues of two different matrices. This is done by…
We define and compute a cohomology of the space of Jacobi forms based on precise analogues of Zhu reduction formulas. A counterpart of the Bott-Segal theorem for the reduction cohomology of Jacobi forms on the torus is proven. It is shown…
We show that every tropical totally positive matrix can be uniquely represented as the transfer matrix of a canonical totally connected weighted planar network. We deduce a uniqueness theorem for the factorization of a tropical totally…
In this paper, a monotonicity property for the quotient of two Jacobi's theta functions with respect to the modulus $k$ is proved.
We prove Cuntz-Krieger and graded uniqueness theorems for Steinberg algebras. We also show that a Steinberg algebra is basically simple if and only if its associated groupoid is both effective and minimal. Finally we use results of…
It is well known that the Jacobi operators completely determine the curvature tensor. The question of existence of a curvature tensor for given Jacobi operators naturally arises, which is considered and solved in the previous work.…
In this paper we give simple extension and uniqueness theorems for restricted additive and logarithmic functional equations.
Spectral averaging techniques for one-dimensional discrete Schroedinger operators are revisited and extended. In particular, simultaneous averaging over several parameters is discussed. Special focus is put on proving lower bounds on the…
We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.
We study the spectral properties of bounded and unbounded Jacobi matrices whose entries are bounded operators on a complex Hilbert space. In particular, we formulate conditions assuring that the spectrum of the studied operators is…
For the periodic matrix-valued Jacobi operator $J$ we obtain the estimate of the Lebesgue measure of the spectrum $|\s(J)|\le4 \min_n\Tr(a_na_n^*)^\frac12$, where $a_n$ are off-diagonal elements of $J$. Moreover estimates of width of…
We consider a one-frequency, quasi-periodic, block Jacobi operator, whose blocks are generic matrix-valued analytic functions. We establish Anderson localization for this type of operator under the assumption that the coupling constant is…
For every $g\geq 2$ we distinguish real period matrices of real Riemann surfaces of topological type $(g,0,0)$ from the ones of topological type $(g,k,1)$, with $k$ equal to one or two for $g$ even or odd respectively (Theorem B). To that…
The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which…
We present in this work a proof of the exponential dichotomy for dynamically defined matrix-valued Jacobi operators in $(\mathbb{C}^{l})^{\mathbb{Z}}$, given for each $\omega \in \Omega$ by the law $[H_{\omega} \textbf{u}]_{n} := D(T^{n -…
We consider $C=A+B$ where $A$ is selfadjoint with a gap $(a,b)$ in its spectrum and $B$ is (relatively) compact. We prove a general result allowing $B$ of indefinite sign and apply it to obtain a $(\delta V)^{d/2}$ bound for perturbations…
This paper explains the fundamental relation between Jacobi structures and the classical Spencer operator coming from the theory of PDEs so as to provide a direct and geometric approach to the integrability of Jacobi structures. It uses…