No abstract

I have examined the nearest-neighbour quantum Heisenberg model on a Cayley tree using a simple density matrix renormalization group scheme. For spin 1/2 on a three-legged Bethe lattice I find indications of long-range antiferromagnetic order in agreement with the spin wave result. Preliminary calculations using a more complicated density matrix renormalization group scheme give promise that one can treat interacting quantum systems on Cayley trees very accurately. † Present and permanent address.

The set P of all probability measures s on the unit circle T splits into three disjoint subsets depending on properties of the derived set of {|j n | 2 ds} n \ 0 , denoted by Lim(s). Here {j n } n \ 0 are orthogonal polynomials in L 2 (ds). The first subset is the set of Rakhmanov measures, i.e., of s ¥ P with {m}=Lim(s), m being the normalized (m(T)=1) Lebesgue measure on T. The second subset Mar(T) consists of Markoff measures, i.e., of s ¥ P with m¨Lim(s), and is in fact the subject of study for the present paper. A measure s, belongs to Mar(T) iff there are e > 0 and l > 0 such that sup{|a n+j |: 0 [ j [ l} > e, n=0, 1, 2, ..., {a n } is the Geronimus parameters (=reflection coefficients) of s. We use this equivalence to describe the asymptotic behavior of the zeros of the corresponding orthogonal polynomials (see Theorem G). The third subset consists of s ¥ P with {m} v Lim(s). We show that s is ratio asymptotic iff either s is a Rakhmanov measure or s satisfies the Ló pez condition (which implies s ¥ Mar(T)). Measures s satisfying Lim(s)={n} (i.e., weakly asymptotic measures) are also classified. Either n is the sum of equal point masses placed at the roots of z n =l, l ¥ T, n=1, 2, ..., or n is the equilibrium measure (with respect to the logarithmic kernel) for the inverse image under an m-preserving endomorphism z Q z n , n=1, 2, ..., of a closed arc J (including J=T) with removed open concentric arc J 0 (including J 0 ="). Next, weakly asymptotic measures are completely described in terms of their Geronimus parameters. Finally, we obtain explicit formulae for the parameters of the equilibrium measures n and show that these measures satisfy {n}=Lim(n).

are non-intersecting closed arcs on the unit circle T then their union E is called rational if all harmonic measures ν E (e j ) at ∞ are rational. It is known that the essential support supp ess (σ ) of a periodic measure σ (i.e. the Verblunsky parameters of σ are periodic) is rational and any rational E is a rotation of supp ess (σ ) for a periodic σ . Elementary proofs of these facts are given. The Schur function f of a periodic σ satisfies z A * f 2where the pair (A, B) of polynomials in z is called a Wall pair for σ . Then supp ess (σ ) = {t ∈ T : |b + (t)| 2 4ω}, b + = B + z B * , ω = C(E) 2 deg(b + ) , C(E) being the logarithmic capacity of E. For any monic b with roots on T, b * = b, and ω satisfying 0 < 4ω m 2 b , where m b is the smallest local maximum of |b| on T, there is a Wall pair (A, B) such that b = B + z B * and supp ess (σ ) = {t ∈ T : |b(t)| 2 4ω} for any periodic σ corresponding to (A, B). The solutions to the equation b = B + z B * in B related to Wall pairs are described. As a consequence we obtain the inverse Bernstein inequality for a separable polynomial b with roots on T: inf T |b | 0.5 · m b · deg(b). The inequality is precise. A complete description of essential supports of periodic measures is also given in terms of the phases of Akhiezer's multi-valued analytic function as well as separable monic polynomials related to it with roots on T.

The convergence in L 2 (T) of the even approximants of the Wall continued fractions is extended to the Cesàro-Nevai class CN, which is defined as the class of probability measures s with lim n Q . 1 n ; n − 1 k=0 |a k |=0, {a n } n \ 0 being the Geronimus parameters of s. We show that CN contains universal measures, that is, probability measures for which the sequence {|j n | 2 ds} n \ 0 is dense in the set of all probability measures equipped with the weak-* topology. We also consider the ''opposite'' Szegő class which consists of measures with ;.1/2 < . and describe it in terms of Hessenberg matrices. © 2002 Elsevier Science (USA)