right limits and reflectionless measures for cmv matrices we study cmv matrices by focusing on their right limit sets we prove a cmv version of a recent result of remling dealing with the implications of the existence of absolutely continuous spectrum and we study some of its consequences we further demonstrate the usefulness of right limits in the study of weak asymptotic convergence of spectral measures and ratio asymptotics for orthogonal polynomials by extending and refining earlier results of khrushchev to demonstrate the analogy with the jacobi case we recover corresponding previous results of simon using the same approach

the absolutely continuous spectrum of jacobi matrices i explore some consequences of a groundbreaking result of brei messer and pearson on the absolutely continuous spectrum of one dimensional schrodinger operators these include an oracle theorem that predicts the potential and rather general results on the approach to certain limit potentials in particular we prove a denisov rakhmanov type theorem for the general nite gap case the main theme is the following it is extremely dicult to produce abso lutely continuous spectrum in one space dimension and thus its existence has strong implications

hierarchical bases for elliptic problems linear systems of equations with positive and symmetric matrices often occur in the numerical treatment of linear and nonlinear elliptic boundary value problems if the cg algorithm is used to solve these equations one is able to speed up the convergence by preconditioning the method of preconditioning with hierarchical basis has already been considered for the laplace equation in two space dimensions and for linear conforming elements in the present work this method is generalized to a large class of conforming and nonconforming elements 0 introduction consider elliptic problems in variational form of order 2m let x htm 2 ci and q w be a bounded domain we want to find u x such that 0 1 a u v i a vmu vmv f v vv x ja here a denotes a strictly positive possibly x dependent matrix f is a functional on x and a is assumed to satisfy c0lmlm 2 o colmlm 2 0 for all u x the solution of this problem exists according to the lax milgram theorem one way to solve this problem numerically is to approximate x by a sequence of finite dimensional spaces xn for a en if ui i xyn is a basis of xn writing u in the form

reflectionless sturm liouville equations we consider compactly supported perturbations of periodic sturm liouville equations in this context one can use the floquet solutions of the periodic background to define scattering coefficients we prove that if the reflection coefficient is identically zero then the operators corresponding to the periodic and perturbed equations respectively are unitarily equivalent in some appendices we also provide the proofs of several basic estimates e g bounds and asymptotics for the relevant m functions

bpx type preconditioners for second and fourth order elliptic problems on the sphere we develop two bramble pasciak xu type preconditioners for second resp fourth order elliptic problems on the surface of the two sphere to discretize the second order problem we construct c 0 linear elements on the sphere and for the fourth order problem we construct c 1 finite elements of powell sabin type on the sphere the main reason these bpx preconditioners work depends on this particular choice of basis we prove optimality and provide numerical examples furthermore we numerically compare the bpx preconditioners with the suboptimal hierarchical basis preconditioners

a topological characterisation of hyperbolic groups we characterise word hyperbolic groups as those groups which act properly discontinuously and cocompactly on the space of distinct triples of a compact metrisable space this is in turn equivalent to a convergence group for which every point of the space is a conical limit point

on the inverse scattering problem for jacobi matrices with the spectrum on an interval a finite system of intervals or a cantor set of positive length when solving the inverse scattering problem for a discrete sturm liouville operator with a rapidly decreasing potential one gets reflection coefficients s and invertible operators where is the hankel operator related to the symbol s the marchenko faddeev theorem 8 in the continuous case for the discrete case see 4 6 guarantees the uniqueness of the solution of the inverse scattering problem in this article we ask the following natural question can one find a precise condition guaranteeing that the inverse scattering problem is uniquely solvable and that operators are invertible can one claim that uniqueness implies invertibility or vise versa moreover we are interested here not only in the case of decreasing potential but also in the case of asymptotically almost periodic potentials so we merge here two mostly developed cases of the inverse problem for sturm liouville operators the inverse problem with almost periodic potential and the inverse problem with the fast decreasing potential

uniformization of sierpinski carpets in the plane let si i i be a countable collection of jordan curves in the extended complex plane widehat mathbb c that bound pairwise disjoint closed jordan regions if the jordan curves are uniform quasicircles and are uniformly relatively separated then there exists a quasiconformal map f colon widehat mathbb c rightarrow widehat mathbb c such that f si is a round circle for all i i this implies that every sierpinski carpet in widehat mathbb c whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round sierpinski carpet by a quasisymmetric map

on isospectral sets of jacobi operators we consider the inverse spectral problem for a class of reflectionless bounded jacobi operators with empty singularly continuous spectra our spectral hypotheses admit countably many accumulation points in the set of eigenvalues as well as in the set of boundary points of intervals of absolutely continuous spectrum the corresponding isospectral set of jacobi operators is explicitly determined in terms of dirichlet type data

preconditioners for nonconforming discretizations we prove an abstract norm equivalence for a two level method which allows us to reduce the construction of preconditioners for nonconforming finite element discretizations to known cases of conforming elements

c 1 hierarchical bases abstract in this note we derive estimates for the condition numbers of stiffness matrices relative to certain c1 conforming hierarchical bases for fourth order elliptic boundary value problems

asymptotics of chebyshev polynomials ii dct subsets of mathbb r we prove szego double acute widom asymptotics for the chebyshev polynomials of a compact subset of which is regular for potential theory and obeys the parreau widom and dct conditions

kotani last problem and hardy spaces on surfaces of widom type this is a small theory of non almost periodic ergodic families of jacobi matrices with purely however absolutely continuous spectrum the reason why this effect may happen is that under our axioms we found an analytic condition on the resolvent set that is responsible for exactly equivalent to this effect

on resonances and the formation of gaps in the spectrum of quasi periodic schrodinger equations we consider one dimensional difference schrodinger equations h x n n 1 n 1 v x n n e n n z x 0 1 with real analytic potential function v x if l e w o is greater than 0 for all e e e and some diophantine 0 then the integrated density of states is absolutely continuous for almost every close to 0 as shown by the authors in earlier work in this paper we establish the formation of a dense set of gaps in spec h x w e e our approach is based on an induction on scales argument and is therefore both constructive as well as quantitative resonances between eigenfunctions of one scale lead to pre gaps at a larger scale to pass to actual gaps in the spectrum we show that these pre gaps cannot be filled more than a finite and uniformly bounded number of times to accomplish this one relates a pre gap to pairs of complex zeros of the dirichlet determinants off the unit circle amongst other things we establish a nonperturbative version of the co variant parametrization of the eigenvalues and eigenfunctions via the phases in the spirit of sinai s perturbative description of the spectrum via his function a this allows us to relate the gaps in the spectrum with the graphs of the eigenvalues parametrized by the phase our infinite volume theorems hold for all diophantine frequencies up to a set of hausdorff dimension zero

two preconditioners based on the multi level splitting of finite element spaces the hierarchical basis preconditioner and the recent preconditioner of bramble pasciak and xu are derived and analyzed within a joint framework this discussion elucidates the close relationship between both methods special care is devoted to highly nonuniform meshes exclusively local properties like the shape regularity of the finite elements are utilized

refinable c2 piecewise quintic polynomials on powell sabin 12 triangulations we present a construction of nested spaces of c 2 macro elements of degree 5 on triangulations of a polygonal domain obtained by uniform refinements of an initial triangulation and a powell sabin 12 split

on multi level bases for elliptic boundary value problems abstract we study the multi level method for preconditioning a linear system arising from a galerkin discretization method of an elliptic boundary value problem of order 2r the solution is approximated in the spline space s10 n when r 1 and s3r 1r 1 n or s 3r 3 r 1 n when r 2 where sd denotes a spline space of smoothness and degree d n is the nth either uniform or nonuniform refinement of a given triangulation 0 and n is the triangulation obtained from the nth refinement either uniform or nonuniform of a given quadrangulation we show that we can construct a multi level basis in these spline spaces which preconditions the linear system so that its condition number is o n 1 2

on the kotani last and schrodinger conjectures in the theory of ergodic one dimensional schrodinger operators ac spectrum has been traditionally expected to be very rigid two key conjectures in this direction state on one hand that ac spectrum demands almost periodicity of the potential and on the other hand that the eigenfunctions are almost surely bounded in the essential suport of the ac spectrum we show how the repeated slow deformation of periodic potentials can be used to break rigidity and disprove both conjectures

the ten martini problem we prove the conjecture known as the ten martini problem after kac and simon that the spectrum of the almost mathieu operator is a cantor set for all nonzero values of the coupling and all irrational frequencies

interpolation of besov spaces we investigate besov spaces and their connection with dyadic spline approximation in lp q 0 p oo our main results are the determination of the interpolation spaces between a pair of besov spaces an atomic decomposition for functions in a besov space the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines

asymptotics of chebyshev polynomials i subsets of r we consider chebyshev polynomials t n z for infinite compact sets e that is the monic polynomials minimizing the sup norm t n e on e we resolve a 45 year old conjecture of widom that for finite gap subsets of r his conjectured asymptotics which we call szeg widom asymptotics holds we also prove the first upper bounds of the form t n e qc e n where c e is the logarithmic capacity of e for a class of e s with an infinite number of components explicitly for those e that obey a parreau widom condition

perturbations of orthogonal polynomials with periodic recursion coefficients the results of denisov rakhmanov szeg shohat nevai and killip simon are extended from asymptotically constant orthogonal polynomials on the real line oprl and unit circle opuc to asymptotically periodic oprl and opuc the key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations

counterexamples to the kotani last conjecture for continuum schrodinger operators via character automorphic hardy spaces abstract the kotani last conjecture states that every ergodic operator in one space dimension with non empty absolutely continuous spectrum must have almost periodic coefficients this statement makes sense in a variety of settings for example discrete schrodinger operators jacobi matrices cmv matrices and continuum schrodinger operators in the main body of this paper we show how to construct counterexamples to the kotani last conjecture for continuum schrodinger operators by adapting the approach developed by volberg and yuditskii to construct counterexamples to the kotani last conjecture for jacobi matrices this approach relates the reflectionless operators associated with the prescribed spectrum to a family of character automorphic hardy spaces and then relates the shift action on the space of operators to the resulting action on the associated characters the key to our approach is an explicit correspondence between the space of continuum reflectionless schrodinger operators associated with a given set and the space of reflectionless jacobi matrices associated with a derived set once this correspondence is established we can rely to a large extent on the previous work of volberg and yuditskii to produce the resulting action on the space of characters we analyze this action and identify situations where we can observe absolute continuity without almost periodicity in the appendix we show how to implement this strategy and obtain analogous results for extended cmv matrices

rigidity for quasi mobius group actions if a group acts by uniformly quasi mobius homeomorphisms on a compact ahlfors n regular space of topological dimension n such that the induced action on the space of distinct triples is cocompact then the action is quasisymmetrically conjugate to an action on the standard n sphere by mobius transformations

hierarchical conforming finite element methods for the biharmonic equation the paper deals with hierarchical bases in spaces of conforming c 1 elements in connection with the approximate solution of the biharmonic equation delta 2 u f quad text in omega qquad u frac partial u partial n 0 quad text on partial omega on a plane polygonal domain omega two different composite finite elements are studied piecewise quadratic powell sabin elements and piecewise cubic elements of clough tocher type the main result are estimates for the condition numbers of the corresponding discretization matrices that show that a conjugate gradient method applied to the hierarchical discretization the so called hierarchical multilevel method will yield suboptimal convergence rates in comparison with standard multigrid schemes

sum rules for jacobi matrices and their applications to spectral theory we discuss the proof of and systematic application of case s sum rules for jacobi matrices of special interest is a linear combination of two of his sum rules which has strictly positive terms among our results are a complete classification of the spectral measures of all jacobi matrices j for which j j 0 is hilbert schmidt and a proof of nevai s conjecture that the szego condition holds if j j 0 is trace class

the boundary of negatively curved groups gromov s article gr contains fundamental properties of negatively curved groups several sets of seminar notes are available frn swn usn that contain more detailed accounts of and further expand on gromov s article a sequence xi e r i 1 2 is convergent at infinity provided xi x t 00 as i j t 00 two sequences xi yj convergent at infinityare equivalent if xi yi t 00 as i t 00 a point in the boundary ar of r is an equivalence class of sequences convergent at infinity now r u ar has a natural topology in which r is a discrete subspace and a typical not necessarily open neighborhood of a point a e ar represented by a sequence xj is given by y e r u arj a y n where a y lim xi y if y e r and a y lim xi yi if year is represented by a sequence yj ar is a compact metrizable finite dimensional space gr swn the important tool that relates ar with the cohomological properties of r is the rips complex pd r for every d 0 pd r is the simplicial complex whose vertices are elements of r and a collection xi xk e r spans a simplex if d xi x d for all i j the natural group action of r on itself by left translations gives rise to an action on pd r the key observation of rips is that when d is sufficiently large pd r is contractible and therefore when r is torsion free provides a model for er more precisely we have proposition 1 1 gr lemma 1 7 a swn 4 2 proposition 9 suppose that r is a negatively curved group and let t5 be as in the definition choose an integer

equality of the spectral and dynamical definitions of reflection for full line jacobi matrices schrodinger operators and cmv matrices we show that being reflectionless in the sense of the well known property of m functions is equivalent to a lack of reflection in the dynamics in the sense that any state that goes entirely to x as t goes entirely to x as t this allows us to settle a conjecture of deift and simon from 1983 regarding ergodic jacobi matrices

reflectionless herglotz functions and jacobi matrices we study several related aspects of reflectionless jacobi matrices first we discuss the singular part of the corresponding spectral measures we then show how to identify sets on which measures are reflectionless by looking at the logarithmic potentials of these measures

hyperbolic groups with planar boundaries we prove that the class of convex cocompact kleinian groups is quasi isometrically rigid we also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually a convex cocompact kleinian group provided that its boundary has ahlfors regular conformal dimension strictly less than 2 or if it acts geometrically on a cat 0 cube complex

spectral structure of anderson type hamiltonians we study self adjoint operators of the form h h 0 n n where the n s are a family of orthonormal vectors and the n s are independently distributed random variables with absolutely continuous probability distributions we prove a general structural theorem saying that for each pair n m if the cyclic subspaces corresponding to the vectors n and m are not completely orthogonal then the restrictions of h to these subspaces are unitarily equivalent with probability one this has some consequences for the spectral theory of such operators in particular we show that well behaved absolutely continuous spectrum of anderson type hamiltonians must be pure and use this to prove the purity of absolutely continuous spectrum in some concrete cases

a characterization of cocompact hyperbolic and finite volume hyperbolic groups in dimension three we show that a cocompact hyperbolic group in dimension 3 is char acterized by certain properties of its word metric which depend only on the group structure and not on any action on hyperbolic space we prove a similar theorem for finite volume hyperbolic groups in dimension 3 a group g is called cocompact hyperbolic of dimension n if and only if it can act isometrically properly discontinuously and cocompactly on hyperbolic space h proper discontinuity implies that the action has at most finite kernel our main result states that a group g is cocompact hyperbolic of dimension 3 if and only if it has a finite generating set c and associated cayley graph f r cr c such that the word metric on t is comparable in the large quasi isometric with the standard hyperbolic metric on h3 we have a similar result for finite volume hyperbolic groups in dimension 3 we believe that our results generalize to all dimensions at least in the cocompact case our result for groups of finite volume employs a different metric space instead of the cayley graph with word metric we use an augmented cayley graph associated with a group g and a finite family of euclidean subgroups of g we prove that the group acts isometrically and properly discontinuously on h3 with orbit space of finite volume if and only if the augmented cayley graph is quasi isometric with h3 precise definitions of all of these terms will appear in 2 the proof of the cocompact case appears in 3 the proof of the finite volume case appears in 4 our results are similar to a result of gromov gl but without the added assumption that the group be the fundamental group of a convex path space there are other results on the classification of groups up to quasi isometry in g2 in particular a group is quasi isometric to e if and only if there is a subgroup z of finite index it would be interesting to know which groups are quasi isometric to the 3 dimensional geometries h2 x e1 solv and psl 2 r gromov s work on groups with polynomial growth g3 bk implies that a group is quasi isometric to nil if and only if it has a subgroup of finite index which is the fundamental group of a compact nil 3 orbifold notice that all compact

c 1 hierarchical riesz bases of lagrange type on powell sabin triangulations in this paper we construct c1 continuous piecewise quadratic hierarchical bases on powell sabin triangulations of arbitrary polygonal domains in r2 our bases are of lagrange type instead of the usual hermite type and under some weak regularity assumptions on the underlying triangulations we prove that they form strongly stable riesz bases for the sobolev spaces hs with s 1 5 2 especially the case s 2 is of interest because we can use the corresponding hierarchical basis for preconditioning fourth order elliptic equations leading to uniformly well conditioned stiffness matrices compared to the hierarchical riesz bases by davydov and stevenson hierarchical riesz bases for hs 1 s 5 2 constructive approximation to appear our construction is simpler

a degree by degree recursive construction of hermite spline interpolants based on the classical hermite spline interpolant h 2 n 1 which is the piecewise interpolation polynomial of class c n 1 and degree 2n 1 a piecewise interpolation polynomial h 2 n of degree 2n is given the formulas for computing h 2 n by h 2 n 1 and computing h 2 n 1 by h 2 n are shown thus a simple recursive method for the construction of the piecewise interpolation polynomial set h j is presented the piecewise interpolation polynomial h 2 n satisfies the same interpolation conditions as the interpolant h 2 n 1 and is an optimal approximation of the interpolant h 2 n 1 some interesting properties are also proved

rigidity of schottky sets we call the complement of a union of at least three disjoint round open balls in the unit sphere bbb s n a schottky set we prove that every quasisymmetric homeomorphism of a schottky set of spherical measure zero to another schottky set is the restriction of a m obius transformation on bbb s n in the other direction we show that every schottky set in bbb s 2 of positive measure admits nontrivial quasisymmetric maps to other schottky sets these results are applied to establish rigidity statements for convex subsets of hyperbolic space that have totally geodesic boundaries

the trace formula for schrodinger operators on the line this paper discusses certain aspects of the spectral and inverse spectral problems for the schrodinger operator l q frac d 2 dx 2 q x for q x c the space of bounded continuous functions the trace formula of the title is the relation q 0 lambda 0 sum limits j 1 infty lambda 2j lambda 2j 1 2 mu j with appropriate choices of j j 0 and j j 1 which is a familiar relation in the theory of hill s equation we characterize the set subseteq c of potentials for which this holds further extensions of the theory of hill s equation are also obtained from the spectrum l q a torust q is constructed which is in general infinite dimensjional everyq x can be mapped to a continuous path ont q described by the auxiliary spectrum j j 1 under certain geometrical conditions on l q this path is the orbit of ac 1 vector field ont q and the mapping extends to one from the hull q overline q x xi xi in mathbb r to the closure of this orbit in particular q is compact these results have applications in the theory of schrodinger operators with ergodic potentials

probabilistic averages of jacobi operators i study the lyapunov exponent and the integrated density of states for general jacobi operators the main result is that questions about these can be reduced to questions about ergodic jacobi operators i use this to show that for finite gap jacobi operators regularity implies that they are in the cesaro nevai class proving a conjecture of barry simon furthermore i use this to study jacobi operators with coefficients a n 1 and b n f n mod 1 for 0 not an integer

absolute continuity of the integrated density of states for the almost mathieu operator with non critical coupling we show that the integrated density of states of the almost mathieu operator is absolutely continuous if and only if the coupling is non critical we deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase settling the measure theoretical case of problem 6 of barry simon s list of schrodinger operator problems for the twenty first century

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

a recursive construction of hermite spline interpolants and applications let f k be the hermite spline interpolant of class c k and degree 2k 1 to a real function f which is defined by its values and derivatives up to order k at some knots of an interval a b we present a quite simple recursive method for the construction of f k we show that if at the step k the values of the kth derivative of f are known then f k can be obtained as a sum of f k 1 and of a particular spline g k 1 of class c k 1 and degree 2k 1 beyond the simplicity of the evaluation of g k 1 we prove that it has other interesting properties we also give some applications of this method in numerical approximation

almost periodicity of some random potentials in this paper we study some random potentials with purely absolutely continuous spectrum in one dimension there are no examples of random potentials with absolutely continuous spectrum that are not almost periodic though it is known from kotani 3 that such potentials are deterministic hence one could conjecture that any random potential with purely absolutely continuous spectrum is almost periodic we show this result for random potentials with infinitely many bands in their spectrum such that the gaps behave well in the finite bands case concini and johnson l discussed the same problem a precise definition of the behavior of the gaps could be found in condition 2 3 we make use of the results of levitan 7 kotani 3 4 and moser 9 for doing this we also discuss the behavior of the lyapunov exponent asymptotically and its relation to the smoothness of the potentials

almost periodic schrodinger operators iii the absolutely continuous spectrum in one dimension we studyh d2 dx2 v x withv x limit periodic e g v x ancos x 2n with an we prove that for a genericv and for generican in the explicit example h is a cantor nowhere dense perfect set for a dense set the spectrum is both cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous

quasisymmetric parametrizations of two dimensional metric spheres according to the classical uniformization theorem every smooth riemannian surface z homeomorphic to the 2 sphere is conformally diffeomorphic to s2 the unit sphere inr3 equipped with the riemannian metric induced by the ambient euclidean metric the availability of a similar uniformization procedure for spheres with a generalized conformal structure is highly desirable in particular in connection with thurston s hyperbolization conjecture this was addressed by cannon in his combinatorial riemann mapping theorem 7 he considers topological surfaces equipped with a sequence of shinglings a combinatorial structure that leads to a notion of approximate conformal moduli of rings he then finds conditions that imply the existence of coordinate systems on the surface that relate these combinatorial moduli to classical analytic moduli in the plane in this paper we develop a uniformization theory for a different type of generalized conformal structure we start with a metric space z homeomorphic to s2 and ask for conditions under which z can be mapped onto s2 by a quasisymmetric homeomorphism the class of quasisymmetries is an appropriate analog of conformal1 mappings in a metric space context quasisymmetric homeomorphisms also arise in the theory of gromov hyperbolic metric spaces quasi isometries between gromov hyperbolic spaces induce quasisymmetric boundary homeomorphisms our setup has the advantage that we can exploit recent notions and methods from analysis

a boundary criterion for cubulation we give a criterion in terms of the boundary for the existence of a proper cocompact action of a word hyperbolic group on a rm cat 0 cube complex we describe applications towards lattices and hyperbolic 3 manifold groups in particular by combining the theory of special cube complexes the surface subgroup result of kahn markovic and agol s criterion we find that every subgroup separable closed hyperbolic 3 manifold is virtually fibered