upper bounds on wavepacket spreading for random jacobi matrices a method is presented for proving upper bounds on the moments of the position operator when the dynamics of quantum wavepackets is governed by a random possibly correlated jacobi matrix as an application one obtains sharp upper bounds on the diffusion exponents for random polymer models coinciding with the lower bounds obtained in a prior work the second application is an elementary argument not using multiscale analysis or the aizenman molchanov method showing that under the condition of uniformly positive lyapunov exponents the moments of the position operator grow at most logarithmically in time

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

on inverse spectral theory for self adjoint extensions mixed types of spectra abstract lethbe a symmetric operator in a separable hilbert space h suppose thathhas some gapj we shall investigate the question about what spectral properties the self adjoint extensions ofhcan have inside the gapjand provide methods on how to construct self adjoint extensions ofhwith prescribed spectral properties insidej under some weak assumptions about the operatorhwhich are satisfied e g provided the deficiency indices ofhare infinite and the operator h 1is compact for one regular point ofh we shall show that for every auxiliary self adjoint operatorm in the hilbert space h and every open subsetj0of the gapjofhthere exists a self adjoint extensionhofhsuch that insidejthe self adjoint extensionhofhhas the same absolutely continuous and the same point spectrum as the given operatorm and the singular continuous spectrum ofhinjequals the closure ofj0inj moreover we shall present a method of how to construct such a self adjoint extensionh via our methods it is possible to construct new kinds of self adjoint realizations of the laplacian on a bounded domain in r d d 1 with spectral properties very different from the spectral properties of the self adjoint realizations known before mathematics subject classification 1991 47a10 47a60 47b25 47e05 47f05

eigenfunctions transfer matrices and absolutely continuous spectrum of one dimensional schrodinger operators on l r and its half line problem h on l2 0 with u 0 0 boundary conditions we will focus on a new approach to the absolutely continuous spectrum ac h and more generally ac h the essential support of the a c part of the spectral measures what is new in our approach is that it relies on estimates on the transfer matrix that is the 2 2 matrix te n m which takes u m 1 u m to u n 1 u n for solutions u of hu eu in the continuum case use u x u x instead of u x 1 u x we let te n te n 0 for example we will prove the following

a theory of sequentiality abstract we show that the notion of sequentiality as presented in concrete data structures may be carried to a more general domain theoretic framework cells are replaced by some linear maps on a domain which turns out to be a di domain in a first phase we do not require the cells to be enabled as they are in cdss and we get a weak form of cartesian closedness then by enriching the structure we get the standard one

polynomials and operator orderings it is shown that there is an exact one to one correspondence between all possible sets of polynomials both orthogonal and nonorthogonal and rules for operator orderings operator orderings that are hermitian give polynomials with definite parity most of the standard classical orthogonal polynomials are associated with operator orderings that are not particularly simple however there is a special one parameter class of hermitian operator orderings that corresponds to a class of elegant but little known orthogonal polynomials called continuous hahn polynomials

a probabilistic approach to one dimensional schrodinger operators with sparse potentials we consider the one dimensional schrodinger equation with sparse potential v i e mainly v 0 it is shown that the asymptotics of the solutions corresponding to positive energies e can be approximately described by an infinite sum of independent random variables using results from probability theory we can then determine the spectral properties of the operators under consideration we prove absolute continuity for a general class of potentials and we also have examples with singular continuous spectrum

p dissipative operator in this paper the authors prove that the generalized positive p selfadjoint gpps operators in banach space satisfy the generalized schwarz inequality solve the maximal dissipative extension representation of p dissipative operators in banach space by using the inequality and introducing the generalized indefinite inner product giip space and apply the result to a certain type of schrodinger operator

euler mahonian statistics via polyhedral geometry abstract a variety of descent and major index statistics have been defined for symmetric groups hyperoctahedral groups and their generalizations typically associated to a pair of such statistics is an euler mahonian distribution a bivariate polynomial encoding the statistics such distributions often appear in rational bivariate generating function identities we use techniques from polyhedral geometry to establish new multivariate identities generalizing those giving rise to many of the known euler mahonian distributions the original bivariate identities are then specializations of these multivariate identities as a consequence of these new techniques we obtain bijective proofs of the equivalence of the bivariate distributions for various pairs of statistics

meixner pollaczek polynomials and the heisenberg algebra an alternative proof is given for the connection between a system of continuous hahn polynomials and identities for symmetric elements in the heisenberg algebra which was first observed by bender mead and pinsky phys rev lett 56 2445 1986 j math phys 28 509 1987 the continuous hahn polynomials turn out to be meixner pollaczek polynomials use is made of the connection between laguerre polynomials and meixner pollaczek polynomials the rodrigues formula for laguerre polynomials an operational formula involving meixner pollaczek polynomials and the schrodinger model for the irreducible unitary representations of the three dimensional heisenberg group

admissible representations of effective cpo s abstract the elements of a cpo complete partial order d are abstract objects in general a concrete machine cannot operate with abstract objects but only with names of objects in this paper the set f of total functions from n to n is suggested as the set of names admissible representations f d for effective cpo s are defined by two axioms which generalize the axioms for godel numberings of the partial recursive functions topological and recursion theoretic considerations show that the definition is very natural it is also proved that p the set of subsets of n is not as suitable as a set of names finally it is proved that for admissible representations the computable functions over cpo s are represented by the computable extensional functions over the names

the lyapunov exponents for schrodinger operators with slowly oscillating potentials abstract by studying the integrated density of states we prove the existence of lyapunov exponents and the thouless formula for the schrodinger operator d 2 dx 2 cos x with 0 l 2 0 this yields an explicit formula for these lyapunov exponents by applying rank one perturbation theory we also obtain some spectral consequences

a stable programming language it is well known that stable models as di domains qualitative domains and coherence spaces are not fully abstract for the language pcf this fact is related to the existence of stable parallel functions and of stable functions that are not monotone with respect to the extensional order which cannot be defined by programs of pcf in this paper a paradigmatic programming language named st pcf is proposed which extends the language pcf with two additional operators the operational description of the extended language is presented in an effective way although the evaluation of one of the new operators cannot be formalized in a pcf like rewrite system since st pcf can define all finite cliques of coherence spaces the above gap with stable models is filled consequently stable models are fully abstract for the extended language

on subordinacy and analysis of the spectrum of schrodinger operators with two singular endpoints the theory of subordinacy is extended to all one dimensional schrodinger operatorsfor which the corresponding differential expression l d 2 dr 2 v r is in the limit point case at both ends of an interval a b with v r locally integrable this enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions of lu xu x e as r a and r b the result provides a rigorous but straightforward method of direct spectral analysis which has very general application and somefurther properties of the spectrum are deduced from the underlying theory

singular continuous spectrum is generic in a variety of contexts we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators those with singular spectrum are a dense g delta

orthogonal polynomials and operator orderings an alternative and combinatorial proof is given for a connection between a system of hahn polynomials and identities for symmetric elements in the heisenberg algebra which was first observed by bender et al resolution of the operator ordering problem by the method of finite elements phys rev lett 56 2445 1986 continuous hahn polynomials and the heisenberg algebra j math phys 28 509 1987 and proven by koornwinder meixner pollaczek polynomials and the heisenberg algebra j math phys 30 767 1989 in the same vein two results announced by bender and dunne polynomials and operator orderings j math phys 29 1727 1988 connecting a special one parameter class of hermitian operator orderings and the continuous hahn polynomials are also proven

pure point spectrum under 1 parameter perturbations and instability of anderson localization we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic fora we study the set of all eigenvalues of the operatora t a tp t that belong to its essential spectrum which does not depend on the parametert we prove that this set is empty for a dense set of values oft then we apply this result or its idea to questions of anderson localization for 1 dimensional schrodinger operators discrete and continuous

singular continuous spectrum under rank one perturbations and localization for random hamiltonians we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic for a in terms of the spectral measure d a we give necessary and sufficient conditions for a p to have empty singular continuous spectrum or to have only point spectrum for a e we apply these results to questions of localization in the one and multi dimensional anderson models

special nonuniform lattice snul orthogonal polynomials on discrete dense sets of points difference calculus compatible with polynomials i e such that the divided difference operator of first order applied to any polynomial must yield a polynomial of lower degree can only be made on special lattices well known in contemporary q calculus orthogonal polynomials satisfying difference relations on such lattices are presented in particular lattices which are dense on intervals q 1 are considered

the system f of variable types fifteen years later the semantic study of system f stumbles on the problem of variable types for which there was no convincing interpretation we develop here a semantics based on the category theoretic idea of direct limit so that the behaviour of a variable type on any domain is determined by its behaviour on finite ones thus getting rid of the circularity of variable types to do so one has first to simplify somehow the extant semantic ideas replacing scott domains by the simpler and more finitary qualitative domains the interpretation obtained is extremely compact as shown on simple examples the paper also contains the definitions of a very small universal model of lambda calculus and investigates the concept totality

on subordinacy and analysis of the spectrum of one dimensional schrodinger operators abstract by considering the behaviour as n of the ratio of l2 0 n norms of solutions of d 2 dr 2 v r u xu 0 r x r a characterisation of the absolutely continuous and singular spectra of one dimensional schrodinger operators is deduced the analysis is applicable to all operators for which l d 2 dr 2 v r is regular at 0 and in the limit point case at infinity with v r locally integrable

operators with singular continuous spectrum i general operators the baire category theorem implies that the family f of dense sets g in fixed metric space x is a candidate for generic sets since it is closed under countable intersections and if x is perfect has no isolated point then a f has uncountable intersections with any open ball in x

heisenberg weyl algebra revisited combinatorics of words and paths the heisenberg weyl algebra which underlies virtually all physical representations of quantum theory is considered from the combinatorial point of view we provide a concrete model of the algebra in terms of paths on a lattice with some decomposition rules we also discuss the rook problem on the associated ferrers board this is related to the calculus in the normally ordered basis from this starting point we explore combinatorial underpinning of the heisenberg weyl algebra which offers novel perspectives methods and applications

operators with singular continuous spectrum ii rank one operators for an operator a with cyclic vector we studya p wherep is the rank one projection onto multiples of if spec a anda has no a c spectrum we prove thata p has purely singular continuous spectrum on for a denseg of s

on the absolutely continuous spectrum of the perturbed hill s equation on considere l equation de hill y q x y y avec x et complexe ou q x est reel continu par morceaux et periodique de periode 1 et l equation augmentee perturbee par une fonction l 1 y q x p x y y ou p x est reel et absolument integrable le long de l axe reel

modified prufer and efgp transforms and deterministic models with dense point spectrum we provide a new proof of the theorem of simon and zhu that in the region e for a e energies d 2 dx 2 cos x 0 1 has lyapunov behavior with a quasi classical formula for the lyapunov exponent we also prove lyapunov behavior for a e e 2 2 for the discrete model with v j 2 e j v n 0 if n 1 4 9 the arguments depend on a direct analysis of the equations for the norm of a solution

boundary conditions of sturm liouville operators with mixed spectra we study bounds on averages of spectral functions corresponding to sturm liouville operators on the half line for different boundary conditions as a consequence constraints are obtained which imply existence of singular spectrum embedded in a c spectrum for sets of boundary conditions with positive measure and potentials vanishing in an interval 0 n these constraints are related to estimates on the measure of sets where the spectral density is positive

trace class perturbations and the absence of absolutely continuous spectra we show that various hamiltonians and jacobi matrices have no absolutely continuous spectrum by showing that under a trace class perturbation they become a direct sum of finite matrices

critical coupling constants and eigenvalue asymptotics of perturbed periodic sturm liouville operators perturbations of asymptotic decay c r 2 arise in the partial wave analysis of rotationally symmetric partial differential operators we show that for each end point 0 of the spectral bands of a perturbed periodic sturm liouville operator there is a critical coupling constant c crit such that eigenvalues in the spectral gap accumulate at 0 if and only if c c crit 1 the oscillation theoretic method used in the proof also yields the asymptotic distribution of the eigenvalues near 0

bounded solutions and absolute continuity of sturm liouville operators this work is a contribution to the theory of the absolutely continuous spectrum of one dimensional schriidinger operators i e sturm liouville operators generated by differential expressions of the form ru u qu in proofs of absolute continuity for such operators and also for the closely related dirac systems always asymptotic properties of solutions of zu lu for real or complex values of 1 have played an important role see for example 18 19 8 4 9 lo in these works and a number of others particular classes of potentials q were treated using properties of solutions in the course of the proof of absolute continuity somewhat more recently the connection between asymptotic properties of solutions and absolute continuity was given a theory i e results of the following form were proved let the solutions of zu lu satisfy some specific asymptotic property for a varying in some subset of r no or almost no a priori assumptions on q then operators generated by z are absolutely continuous in this set a first result of this type was given by carmona 3 who proved that local uniform boundedness of solutions together with their derivatives in some open a set implies pure absolute continuity in this set here we will mostly rely on the concept of non or racy of solutions which was shown by gilbert and pearson 7 12 61 to be a powerful tool in the investigation of the continuous spectrum of sturm liouville operators in section 2 we give some consequences of their results whi ch are useful in proofs of absolute continuity these consequences range from non existence of imbedded eigenvalues and the existence of absolutely continuous parts to pure absolute continuity and the existence of transient 210 0022 247x 92 5 00

operators with singular continuous spectrum iii almost periodic schrodinger operators we prove that one dimensional schrodinger operators with even almost periodic potential have no point spectrum for a dense g in the hull this implies purely singular continuous spectrum for the almost mathieu equation for coupling larger than 2 and a dense g in even if the frequency is an irrational with good diophantine properties

one dimensional perturbations of unbounded selfadjoint operators with empty spectrum abstract we study spectral properties of one dimensional singular nonselfadjoint perturbations of an unbounded selfadjoint operator and give criteria for the possibility to remove the whole spectrum by a perturbation of this type a counterpart of our results for the case of bounded operators provides a complete description of compact selfadjoint operators whose rank one perturbation is a volterra operator

wkb asymptotic behavior of almost all generalized eigenfunctions for one dimensional schrodinger operators with slowly decaying potentials we prove the wkb asymptotic behavior of solutions of the differential equation d2u dx2 v x u eu for a e e a where v v1 v2 v1 lp r and v2 is bounded from above with a lim supx v x while v 2 x lp r 1 p 2 these results imply that schrodinger operators with such potentials have absolutely continuous spectrum on a we also establish wkb asymptotic behavior of solutions for some energy dependent potentials

stability of singular spectral types under decaying perturbations we look at invariance of a e boundary condition spectral behavior under perturbations w of half line continuum or discrete schrodinger operators we extend the results of del rio simon stolz from compactly supported w s to suitable short range w we also discuss invariance of the local hausdorff dimension of spectral measures under such perturbations

spectral analysis on infinite sierpinski gaskets we study the spectral properties of the laplacian on infinite sierpin ski gaskets we prove that the laplacian with the neumann boundary condition has pure point spectrum moreover the set of eigenfunctions with compact support is complete the same is true if the infinite sierpin ski gasket has no boundary but is false for the laplacian with the dirichlet boundary condition in all these cases we describe the spectrum of the laplacian and all the eigenfunctions with compact support to obtain these results first we prove certain new formulas for eigenprojectors of the laplacian on finite sierpin ski pre gaskets then we prove that the spectrum of the discrete laplacian on a sierpin ski lattice is pure point and the eigenfunctions are localized 1998 academic press

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

modified prufer and efgp transforms and the spectral analysis of one dimensional schrodinger operators using control of the growth of the transfer matrices wediscuss the spectral analysis of continuum and discrete half line schrodinger operators with slowly decaying potentials among our results we show if where w has compact support and then h has purely a c resp purely s c spectrum on 0 if for n 1 2 a n potentials where a n are independent identically distributed random variables with e a n 0 e a 2 n 1 and 2 we find singular continuous spectrum with explicitly computable fractional hausdorff dimension

stability and computability in coherent domains abstract the relation between stability and sequentiality is investigated in the category of girard s coheent domains we introduce and discuss a notion of computability for stable functions based on the recursive enumerability of their traces in a way similar to the definition of computable functions in scott s effectively given domains we then relate this notion of computability to regular sets and relative algorithms oracle machines of the theory of relativized computability the notion of oracle machine is used to formalize the idea of a main sequential program which calls an unspecified external agent o a sort of subroutine call in particular we prove that a function f between two coherent domains x and y is stable and computable if and only if f may be computed by an oracle machine questioning in a positive way a simple class of oracles that supply informations about elements in x

absolutely continuous spectrum of one dimensional schrodinger operators and jacobi matrices with slowly decreasing potentials we prove that for any one dimensional schrodinger operator with potentialv x satisfying decay condition v x cx 3 4 e the absolutely continuous spectrum fills the whole positive semi axis the description of the set in on which the singular part of the spectral measure might be supported is also given analogous results hold for jacobi matrices

one dimensional schrodinger operators with random or deterministic potentials new spectral types abstract we prove sufficient conditions involving only potential asymptotic near one of the infinities in order to have purely absolutely continuous components in the spectrum these deterministic results are then applied to random cases and exhibit classes of models for which with probability one one component of the spectrum is purely absolutely continuous and the rest is dense pure point with exponentially decaying eigenfunctions

from sequences to polynomials and back via operator orderings bender and dunne polynomials and operator orderings j math phys 29 1727 1731 1988 showed that linear combinations of words qkpnqn k where p and q are subject to the relation qp pq i may be expressed as a polynomial in the symbol z 12 qp pq relations between such polynomials and linear combinations of the transformed coefficients are explored in particular examples yielding orthogonal polynomials are provided

instability of dense point spectrum under finite rank perturbations we prove that if a is a self adjoint operator and r a non negative finite rank operator whose range is cyclic for a then for a generic in the baire sense t in each eigenvalue of the operator has a neighborhood containing no other eigenvalues

continuous spectrum and uniform localization for ergodic schrodinger operators abstract in this paper we study localization for ergodic families of discrete schrodinger operators we prove that instability of pure point spectrum implies absence of uniform localization

transient and recurrent spectrum abstract we deal primarily with spectral analysis of an abstract self adjoint operator h on a hilbert space h we propose a further refinement of the absolutely continuous subspace h ac into the transient absolutely continuous subspace h tac which is the closure of those with e ith o t n for all n and the recurrent absolutely continuous subspace h rac h ac h tac we discuss general features of this breakup in a subsequent paper we construct analytic almost periodic functions v on so that h d 2 dx 2 v x on l 2 has only recurrent absolutely continuous spectrum in the sense that h rac h

singular continuous spectrum for the period doubling hamiltonian on a set of full measure we consider the discrete one dimensional schrodinger operator with potential generated by the period doubling substitution we show that for almost every element in the hull with respect to the unique ergodic measure on there are no eigenvalues combining this with a result proven by kotani we establish purely singular continuous spectrum on a set of full measure

computability on di domains in this paper we investigate a subset of the class of scott computable stable functions called berry computable furthermore we introduce a corresponding notion of an effectively given di domain we obtain the result that the category with these domains as objects and berry computable functions as morphisms has all the important attributes of recursive domains and scott computable continuous functions e g cartesian closedness moreover we investigate the relationship between stable functions and sequential algorithms we show that any berry computable function can be computed in a sequential way by a relative algorithm additionally if the codomain of the considered stable function is an atomic di domain we obtain the result that the reverse holds as well

operators with singular continuous spectrum vii examples with borderline time decay we construct one dimensional potentials v x so that if h d 2 dx 2 v x on l 2 then h has purely singular spectrum but for a dense set d d implies that ith c t 1 2 in t for t 2 this implies the spectral measures have hausdorff dimension one and also following an idea of malozemov molchanov provides counterexamples to the direct extension of the theorem of simon spencer on one dimensional infinity high barriers

some schrodinger operators with power decaying potentials and pure point spectrum we construct deterministic potentials such that the schrodinger equation on has dense pure point spectrum in for almost all boundary conditions at as a by product we also obtain power decaying potentials for which the spectrum is purely singular continuous on for all boundary conditions