direct and inverse spectral problems for a class of non self adjoint periodic tridiagonal matrices abstract the spectral properties of a class of tridiagonal matrices are investigated the reconstruction of matrices of this special class from given spectral data is also studied necessary and sufficient conditions for that reconstruction are found the obtained results extend some results on the direct and inverse spectral problems for periodic jacobi matrices and for some non self adjoint tridiagonal matrices

contact structures equivariant spin bordism and periodic fundamental groups using contact surgery and equivariant bordism theory we prove the existence of contact structures on all 5 dimensional spin manifolds with certain finite fundamental groups

invariant functionals and the uniqueness of invariant norms let be a representation of a compact group g on a banach space x the question we address is whether x carries a unique invariant norm in the sense that is the unique norm on x for which is a representation we characterize the uniqueness of norm in terms of the automatic continuity of the invariant functionals in the case when x is a dual banach space and is a x x continuous representation of g on x such that g consists of x x continuous operators we illustrate the usefulness of this characterization by studying the uniqueness of the norm on the spaces lp where is a locally compact hausdorff space equipped with a positive radon measure and g acts on as a group of continuous invertible measure preserving transformations

the yamabe invariant for non simply connected manifolds the yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature recently petean showed that the yamabe invariant is nonnegative for all closed simply connected manifolds of dimension 5 we extend this to show that yamabe invariant is nonnegative for all closed manifolds of dimension 5 with fundamental group of odd order having all sylow subgroups abelian the main new geometric input is a way of studying the yamabe invariant on toda brackets a similar method of proof shows that all closed oriented manifolds of dimension 5 with non spin universal cover with finite fundamental group having all sylow subgroups elementary abelian admit metrics of positive scalar curvature once one restricts to the complement of manifolds whose homology classes are toral the exceptional toral homology classes only exist in dimensions not exceeding the rank of the fundamental group so this proves important cases of the gromov lawson rosenberg conjecture once the dimension is sufficiently large

lp uniqueness of schrodinger operators and the capacitary positive improving property abstract we prove several lp uniqueness results for schrodinger operators l v by means of the feynman kac formula using the m p capacity theory for general markov semigroups we show that the associated feynman kac semigroup is positive improving in the sense of m p capacity improving the well known one in the sense of measure using that capacitary positive improving property and two new inequalities for generalized ornstein uhlenbeck generators we show the essential self adjointness of the ground state diffusion generator l l 2 associated with two dimensional euclidean quantum fields

spectral problems for generalized jacobi matrices a new class of generalized jacobi matrices is introduced every proper real rational function is proved to be the m function of a unique finite generalized jacobi matrix moreover every generalized nevanlinna function m which is a solution of a determinate indefinite moment problem turns out to be the m function of a unique infinite generalized jacobi matrix the method we use is based on the step by step schur process of solving the indefinite moment problem the convergence of the sequence of subdiagonal pade approximants for the corresponding hamburger series is investigated

modulus of continuity of the canonic brownian motion on the group of diffeomorphisms of the circle abstract an explicit modulus of holder continuity is given for the flow associated to the canonic brownian motion on the diffeomorphism group of the circle

dirichlet operators a priori estimates and the uniqueness problem abstract we study the equation du dt hu 0 where h is the operator associated with the dirichlet form of quasi invariant measure on hilbert space a priori estimates of the first and second order derivatives of the solutions of the equation are obtained then these estimates are applied to the problem of essential self adjointness of the operator h on domains which consist of smooth functions

uniqueness of l 1 solutions for the laplace equation and the heat equation on riemannian manifolds on deduit une hypothese geometrique optimale sur une variete de riemann complete qui assure des proprietes d unicite pour des solutions l 1 de l equation de laplace et de la chaleur

absolutely continuous spectrum for the isotropic maxwell operator with coefficients that are periodic in some directions and decay in others 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 as in 3 4

topologies on homeomorphism spaces of certain metric spaces a new proof for the equivalence of several topologies on homeomorphism groups over certain metric spaces x is given which is based on the metric of x

on the spectrum of waveguides in planar photonic bandgap structures we study a helmholtz type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides the waveguide is created by introducing a linear defect into a two dimensional periodic medium the defect is infinitely extended and aligned with one of the coordinate axes this perturbation introduces guided mode spectrum inside the band gaps of the fully periodic unperturbed spectral problem in the first part of the paper we prove that guided mode spectrum can be created by arbitrarily small perturbations secondly we show that after performing a floquet decomposition in the axial direction of the waveguide for any fixed value of the quasi momentum kx the perturbation generates at most finitely many new eigenvalues inside the gap

contractions with rank one defect operators and truncated cmv matrices the main issue we address in the present paper are the new models for completely nonunitary contractions with rank one defect operators acting on some hilbert space of dimension n these models complement nicely the well known models of livsic and sz nagy foias we show that each such operator acting on some finite dimensional respectively separable infinite dimensional hilbert space is unitarily equivalent to some finite respectively semi infinite truncated cmv matrix obtained from the full cmv matrix by deleting the first row and the first column and acting in cn respectively l2 n this result can be viewed as a nonunitary version of the famous characterization of unitary operators with a simple spectrum due to cantero moral and velazquez as well as an analog for contraction operators of the result from yu arlinski e tsekanovski non self adjoint jacobi matrices with a rank one imaginary part j funct anal 241 2006 383 438 concerning dissipative non self adjoint operators with a rank one imaginary part it is shown that another functional model for contractions with rank one defect operators takes the form of the compression f pk f on the hilbert space l2 t d with a probability measure onto the subspace k l2 t d c the relationship between characteristic functions of sub matrices of the truncated cmv matrix with rank one defect operators and the corresponding schur iterates is established we develop direct and inverse spectral analysis for finite and semi infinite truncated cmv matrices in particular we study the problem of reconstruction of such matrices from their spectrum or the mixed spectral data involving schur parameters it is pointed out that if the mixed spectral data contains zero eigenvalue then no solution unique solution or infinitely many solutions may occur in the inverse problem for truncated cmv matrices the uniqueness theorem for recovered truncated cmv matrix from the given mixed spectral data is established in this part the paper is closely related to the results of hochstadt and gesztesy simon obtained for finite self adjoint jacobi matrices

uniqueness of translation invariant norms let a be a banach function space and let m be a family of multipliers on a we provide conditions on m so that the original topology of a is the only complete norm topology on a making all of the maps from m continuous as a corollary we show that for a compact abelian group g and a circle group t for a lp t 1 p the lp norm is the only one that makes all translations continuous while for a c g a l g or a l1 g there are other norms with that property for noncompact groups the situation is different on the space l1 r the l1 norm is the only one that makes a single nontrivial translation continuous

opening up and control of spectral gaps of the laplacian in periodic domains the main result of this work is as follows for arbitrary pairwise disjoint finite intervals j j 0 j 1 m and for arbitrary n 2 we construct a family of periodic non compact domains e n e 0 such that the spectrum of the neumann laplacian in e has at least m gaps when e is small enough moreover the first m gaps tend to the intervals j j as e 0 the constructed domain e is obtained by removing from n a system of periodically distributed trap like surfaces the parameter e characterizes the period of the domain e also it is involved in a geometry of the removed surfaces

on the determination of a tridiagonal matrix from its spectrum and a submatrix abstract given a set of 2 n real numbers 1 2 2 n the authors describe the set s of n n tridiagonal matrices with the property that each s can be completed to a 2 n 2 n tridiagonal matrix l with spec l 1 2 2 n

a modified method for reconstructing periodic jacobi matrices in this note we discuss the reconstruction of periodic jacobi matrices from spectral data the method combines ideas and techniques from the algorithms given by boley and golub 1 2 and ferguson 3 resulting in a numerically stable algorithm applicable to a larger class of problems the number of initial data items needed for this method equals the number of items in the resulting matrix namely 2n

on guided waves in 3 d photonic crystals the propagation of guided waves in 3 d photonic crystals is studied in this paper we suppose that the spectrum of reference medium a perfectly 3 d photonic crystal has band gaps if a line defect is introduced along one of the lattice vectors we prove that spectrum can be created inside the gaps of the reference spectrum if the line defect is sufficiently strong furthermore we also prove that the generalized eigenfunctions corresponding to the generalized eigenvalues created by line defect are exponentially decaying away from the line defect

uniqueness of nelsons diffusions let 2 be a generalized schrodinger operator or generator of nelsons diffusion defined on c 0 d where is a continuous and strictly positive function on an open domain d d such that lloc2 d some results are given about the two questions below i whether does generate a unique semigroup in l1 d 2dx ii whether the semigroup determined by is strong feller

inverse spectral theory of finite jacobi matrices we solve the following physically motivated problem to determine all finite jacobi matrices j and corresponding indices i j such that the green s function is proportional to an arbitrary prescribed function f z our approach is via probability distributions and orthogonal polynomials we introduce what we call the auxiliary polynomial of a solution in order to factor the map j i j where square brackets denote the equivalence class consisting of scalar multiples this enables us to construct the solution set as a fibration over a connected semi algebraic coordinate base the end result is a wealth of explicit constructions for jacobi matrices these reveal precise geometric information about the solution set and provide the basis for new existence theorems

eigenvalue branches of the perturbed maxwell operator m λd in a gap of σ m the propagation of guided waves in photonic crystal fibers pcfs is studied a pcf can be regarded as a perfectly two dimensional photonic crystal with a line defect along the axial direction this problem can be treated as an eigenvalue problem for a family of noncompact self adjoint operators under the assumption that the background spectrum has a gap we prove that a line defect can create an eigenvalue of any given fixed value in the gap provided that the defect is strong enough based on a decoupling of regions in r2 by means of dirichlet and neumann boundaries then using the trace ideal estimates we study asymptotic distribution of eigenvalues and bounds on the number of eigenvalue branches in particular we show that if the defect is weak enough no eigenvalues can be created inside the gap

periodic riemannian manifold with preassigned gaps in spectrum of laplace beltrami operator it is known e l green 1997 o post 2003 that for an arbitrary m n one can construct a periodic non compact riemannian manifold m with at least m gaps in the spectrum of the corresponding laplace beltrami operator m in this work we want not only to produce a new type of periodic manifolds with spectral gaps but also to control the edges of these gaps the main result of the paper is as follows for arbitrary pairwise disjoint intervals j j 0 j 1 m m n for an arbitrarily small 0 and for an arbitrarily large l 0 we construct a periodic non compact riemannian manifold m with at least m gaps in the spectrum of the operator m moreover the edges of the first m gaps belong to neighbourhoods of the edges of the intervals j j while the remaining gaps if any are located outside the interval 0 l

structured matrices continued fractions and root localization of polynomials we give a detailed account of various connections between several classes of objects hankel hurwitz toeplitz vandermonde and other structured matrices stietjes and jacobi type continued fractions cauchy indices moment problems total positivity and root localization of univariate polynomials along with a survey of many classical facts we provide a number of new results

continued fractions and periodic jacobi matrices abstract we describe an algorithm based on a continued fraction expansion to reconstruct a periodic jacobi matrix from spectral data we also compare various different sets of spectral data which characterize a periodic jacobi matrix

the discrete spectrum of the periodic maxwell operator perturbed by a decreasing potential the propagation of electromagnetic waves in dielectric medium with decreasing perturbations defects is studied we use the maxwell operator m p x and the perturbed operator m p x q x to describe the background medium and perturbed medium respectively where 0 is the coupling constant under the assumption that m has a spectral gap g we prove that a defect can create an eigenvalue of any given fixed value g for some 0 provided that the support of the defect is large enough we also consider the asymptotic distribution of eigenvalues in the gap and bounds on the number of eigenvalue branches

an overview of periodic elliptic operators 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 influence significantly most properties of such operators the approaches described are applicable not only to the standard model example of schrodinger operator with periodic electric potential v x but to a wide variety of elliptic periodic equations and systems equations on graphs operator and other operators on abelian coverings of compact bases many important applications are mentioned however due to the size restrictions they are not dealt with in details

the image of j in the ehp sequence the ehp sequence is based on a result of james j who showed that there is h a map h such that s q sn s2n 1 is a fibration when localized at 2 the map sn sn is usually labeled e the boundary homomorphism in the homotopy sequence is usually labeled p for our purposes it will be most convenient to combine all the ehp sequences into one system this gives the following filtration of o201 s q s

non self adjoint jacobi matrices with a rank one imaginary part we develop direct and inverse spectral analysis for finite and semi infinite non self adjoint jacobi matrices with a rank one imaginary part it is shown that given a set of n not necessarily distinct nonreal numbers in the open upper lower half plane uniquely determines an n n jacobi matrix with a rank one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity algorithms of reconstruction for such finite jacobi matrices are presented a new model complementing the well known livsic triangular model for bounded linear operators with a rank one imaginary part is obtained it turns out that the model operator is a non self adjoint jacobi matrix we show that any bounded prime non self adjoint linear operator with a rank one imaginary part acting on some finite dimensional respectively separable infinite dimensional hilbert space is unitarily equivalent to a finite respectively semi infinite non self adjoint jacobi matrix this obtained theorem strengthens a classical result of stone established for self adjoint operators with simple spectrum we establish the non self adjoint analogs of the hochstadt and gesztesy simon uniqueness theorems for finite jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non self adjoint tri diagonal matrices to the case of mixed eigenvalues real and nonreal a unique jacobi matrix unitarily equivalent to the operator of integration ff x 2i xlf t dt in the hilbert space l2 0 l is found as well as spectral properties of its perturbations and connections with the well known bernoulli numbers we also give the analytic characterization of the weyl functions of dissipative jacobi matrices with a rank one imaginary part

applications of kazhdan s property t to automatic continuity let g be a compact group which has the strong kazhdan s property t and let be an isometric representation of g on l p for some measure space and 1 p we characterize the continuity of all the linear operators from l p which commute with translations and the property of p being the unique norm up to equivalence on l p with respect to which all translations are continuous in terms of the finite dimensionality of the subspace of all invariant func

eta invariants and their adiabatic limits 0 introduction 1 algebraic preliminaries 2 the finite dimensional case superconnections a odd dimensional base spaces b even dimensional base spaces appendix 1 coupling to general superconnections 3 auxiliary grassman variables and auxiliary adiabatic limits 4 the infinite dimensional case fibrations a elementary geometry of fibrations b odd dimensional base spaces c even dimensional base spaces d the case in which the dirac operator on the fiber is not invertible appendix 2 multifibrations

on the uniqueness problem for dirichlet operators abstract strong uniqueness in l 2 and in l 1 for dirichlet operators in finite dimensional spaces is studied

the classical moment problem as a self adjoint finite difference operator abstract this is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators among the advantages of this approach is that the nevanlinna functions appear as elements of a transfer matrix and convergence of pade approximants appears as the strong resolvent convergence of finite matrix approximations to a jacobi matrix as a bonus of this we obtain new results on the convergence of certain pade approximants for series of hamburger

gaps in the spectrum of the maxwell operator with periodic coefficients the periodic maxwell operator is considered piecewise constant coefficients are constructed in such a way that the spectrum of operator has the gaps

spectral gaps for water waves above a corrugated bottom in this paper the essential spectrum of the linear problem on water waves in a water layer and in a channel with a gently corrugated bottom is studied we show that under a certain geometric condition the essential spectrum has spectral gaps in other words there exist intervals in the positive real semi axis that are free of the spectrum but have their endpoints in it the position and the length of the gaps are found out by applying an asymptotic analysis to the model problem in the periodicity cell

complex jacobi matrices complex jacobi matrices play an important role in the study of asymptotics and zero distribution of formal orthogonal polynomials fops the latter are essential tools in several fields of numerical analysis for instance in the context of iterative methods for solving large systems of linear equations or in the study of pade approximation and jacobi continued fractions in this paper we present some known and some new results on fops in terms of spectral properties of the underlying infinite jacobi matrix with a special emphasis to unbounded recurrence coefficients here we recover several classical results for real jacobi matrices the inverse problem of characterizing properties of the jacobi operator in terms of fops and other solutions of a given three term recurrence is also investigated this enables us to give results on the approximation of the resolvent by inverses of finite sections with applications to the convergence of pade approximants

real connective k theory and the quaternion group let ko be the real connective k theory spectrum we compute ko bg and ko bg for groups g whose sylow 2 subgroup is quaternion of order 8 using this we compute the coefficients t ko g of the g fixed points of the tate spectrum t ko for g s12 3 and g q8 the results provide a counterexample to the optimistic conjecture of greenlees and may 9 conj 13 4 by showing in particular that t ko g is not a wedge of eilenberg mac lane spectra as occurs for groups of prime order

uniqueness theorems in inverse spectral theory for one dimensional schrodinger operators new unique characterization results for the potential v x in connection with schrodinger operators on r and on the half line 0 are proven in terms of appropriate krein spectral shift functions particular results obtained include a generalization of a well known uniqueness theorem of borg and marchenko for schrodinger operators on the half line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for schrodinger operators with confining potentials on the entire real line

the essential self adjointness of generalized schrodinger operators a necessary and sufficient condition is given for the generalized schrodinger operator a 12 i 1n di di to be essentially self adjoint in l2 dx under general assumptions on and for arbitrary domains in rn in particular if is strictly positive and locally lipschitz continuous on rn then a is essentially self adjoint examples of non essential self adjointness and a complete discussion of the one dimensional case are also given these results have applications to the problem of the essential self adjointness of quantum hamiltonians and to the uniqueness problem of markov processes