Absolutely continuous spectrum of a Schr\"odinger operator on a tree

We give sufficient conditions for the presence of the absolutely continuous spectrum of a Schr\"odinger operator on a regular rooted tree without loops (also called regular Bethe lattice or Cayley tree).Comment: 10 pages, 1 figure; a preliminary version, few more typos correcte

Subtraction Division Games

A Subtraction-Division game is a two player combinatorial game with three parameters: a set S, a set D, and a number n. The game starts at n, and is a race to say the number 1. Each player, on their turn, can either move the total to n-s for some s in S or to \lceil\frac{n}{d}\rceil, for some d in D (i.e. either subtract s or divide by d and round up). This paper examines in detail the Sprague-Grundy sequences for the games when |S|=|D|=1. For a large class of these games, the patterns in the sequences allow for a characterization of all the winning positions. The paper also proves that, for the same large class, these sequences are automatic. In the process, a large family of recurrences that completely determine their behavior is constructed

Operators Similar to Contractions and Their Similarity to a Normal Operator

It is proved recently by Benamara-Nikolski that a contraction having finite defects and spectrum not filling in the closed unit disc, is similar to a normal operator if and only if it has the so-called linear resolvent growth property. We obtain results of the same type for a wider (than contractions) class of operators.Comment: LaTex, 17 pages, preliminary versio

On complex perturbations of infinite band Schrodinger operators

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.Comment: 11 pages, 1 figure, submitted to Methods of Functional Analysis and Topolog

Criteria for Similarity of a Dissipative Integral Operator to a Normal Operator

We consider an integral dissipative operator in its Brodskii-Livshits triangular representation. The main question we are concerned with is similarity of the operator to a normal one. We obtain necessary as well as sufficient conditions for the similarity. The study is based on functional model technique.Comment: AMSTex, 31 page

The Szego class with a polynomial weight

Let p be a trigonometric polynomial, nonnegative on the unit circle $\mathbb{T}$. We say that a measure $\sigma$ on $\mathbb{T}$ belongs to the polynomial Szego class, if $d\sigma=sigma'_{ac}d\theta+d\sigma_s$, $\sigma_s$ is singular, and $p\ln \sigma'_{ac}$ is summable on $\mathbb{T}$. For the associated orthogonal polynomials, we obtain pointwise asymptotics inside the unit disc. Then, we show that this asymptotics holds in the $L^2$ sense on the unit circle. As a corollary, we get existense of certain modified wave operators.Comment: preliminary versio

Lieb--Thirring bounds for complex Jacobi matrices

We obtain various versions of classical Lieb--Thirring bounds for one- and multi-dimensional complex Jacobi matrices. Our method is based on Fan-Mirski Lemma and seems to be fairly general.Comment: 10 pages, 1 figure; a preliminary versio

Asymptotics of the orthogonal polynomials for the Szego class with a polynomial weight

Let p(t) be a trigonometric polynomial, non-negative on the unit circle. We say that a measure \sigma belongs to a polynomial Szego class, if the logarithm of its density is summable over the circle with the weight p(t). For the associated orthogonal polynomials, we obtain pointwise asymptotics inside the unit disc. Then, we show that these asymptotics holds in L^2-sense on the unit circle. As a corollary, we get an existence of certain modified wave operators.Comment: revised versio

On the growth of the polynomial entropy integrals for the measures in the Szego class

For the polynomials orthogonal on the unit circle with respect to the measure from the Szego class we prove that the polynomial entropy integrals can grow. The estimate obtained is sharp

On non-selfadjoint perturbations of infinite band Schr\"odinger operators and Kato method

Let H_0=-\dd+V_0 be a multidimensional Schr\"odinger ope\-rator with a real-valued potential and infinite band spectrum, and $H=H_0+V$ be its non-selfadjoint perturbation defined with the help of Kato approach. We prove Lieb--Thirring type inequalities for the discrete spectrum of $H$ in the case when V_0\in L^\infty(\br^d) and V\in L^p(\br^d), $p>\max(d/2, 1)$.Comment: 12 pages, 1 figure. arXiv admin note: text overlap with arXiv:1502.0602