singular density of states measure for subshift and quasi periodic schrodinger operators simon s subshift conjecture states that for every aperiodic minimal subshift of verblunsky coefficients the common essential support of the associated measures has zero lebesgue measure we disprove this conjecture in this paper both in the form stated and in the analogous formulation of it for discrete schrodinger operators in addition we prove a weak version of the conjecture in the schrodinger setting namely under some additional assumptions on the subshift we show that the density of states measure a natural measure associated with the operator family and whose topological support is equal to the spectrum is singular we also consider one frequency quasi periodic schrodinger operators with continuous sampling functions and show that generically the density of states measure is singular as well

invariance of white noise for kdv on the line we consider the korteweg de vries equation with white noise initial data posed on the whole real line and prove the almost sure existence of solutions moreover we show that the solutions obey the group property and follow a white noise law at all times past or future as an offshoot of our methods we also obtain a new proof of the existence of solutions and the invariance of white noise measure in the torus setting

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

the circulant operator in the banach algebra of matrices abstract we study an operator c which maps every n by n matrix a n to a circulant matrix c a n that minimizes the frobenius norm a n c n f over all n by n circulant matrices c n the circulant matrix c a n called the optimal circulant preconditioner has proved to be a good preconditioner for a general class of toeplitz systems in this paper we give different formulations of the operator discuss its algebraic and geometric properties and compute its operator norms in different banach algebras of matrices using these results we are able to give an efficient algorithm for finding the superoptimal circulant preconditioner which is defined to be the minimizer of i c n 1 a n f over all nonsingular circulant matrices c n

the generalized superoptimal preconditioner abstract for any given n by n matrix a a specific circulant preconditioner t f a introduced by tyrtyshnikov e tyrtyshnikov optimal and super optimal circulant preconditioners siam j matrix anal appl 13 1992 459 473 is defined to be the solution of min c i c 1 a f over all n by n nonsingular circulant matrices c the preconditioner t f a called the superoptimal circulant preconditioner has been proved to be a good preconditioner for a large class of structured systems in this paper we study this preconditioner in the general case by using the moore penrose inverse we give a formula for the superoptimal preconditioner and discuss the stability properties of this preconditioner a spectral relation between the optimal and superoptimal preconditioned matrices in the general case is also given

the characterizations of a semicircle law by the certain freeness in a c probability space in usual probability theory various characterizations of the gaussian law have been obtained for instance independence of the sample mean and the sample variance of independently identically distributed random variables characterizes the gaussian law and the property of remaining independent under rotations characterizes the gaussian random variables in this paper we consider the free analogue of such a kind of characterizations replacing independence by freeness we show that freeness of the certain pair of the linear form and the quadratic form in freely identically distributed noncommutative random variables which covers the case for the sample mean and the sample variance characterizes the semicircle law moreover we give the alternative proof for nica s result that the property of remaining free under rotations characterizes a semicircular system our proof is more direct and straightforward one

on the evolution of a reflection coefficient under the korteweg de vries flow we are concerned with the korteweg de vries equation on the full line with real nondecaying initial profiles we find the time evolution of a relative reflection coefficient an inverse spectral formalism is also considered for certain mixed problems on the full line

superconvergence to the central limit and failure of the cramer theorem for free random variables we show that convergence of the semicircle law in the free central limit theorem for bounded random variables is much better than expected thus the distributions which tend to the semicircle become absolutely continuous in finite time and the densities converge in a very strong sense we also show that the semicircle law is the free convolution of laws which are not semicircular thus proving that cramer s classical result for the normal distribution does not have a free counterpart

a note on spectra of optimal and superoptimal preconditioned matrices in this short note it is proved that given any positive definite hermitian matrix the eigenvalues of the superoptimal preconditioned matrix do not exceed the corresponding eigenvalues of the optimal preconditioned matrix

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

zeros of the wronskian and renormalized oscillation theory for general sturm liouville operators with separated boundary conditions we prove the following if e 1 2 r and if u 1 2 solve the differential equation hu j e ju j j 1 2 and respectively satisfy the boundary condition on the left right then the dimension of the spectral projection p e 1 e 2 h of h equals the number of zeros of the wronskian of u 1 and u 2

resolvents of r diagonal operators we consider the resolvent a 1 of any diagonal operator a in a ii 1 factor our main theorem theorem 1 1 gives a universal asymptotic formula for the norm of such a resolvent en route to its proof we calculate the transform of the operator c 2 where c is voiculescu s circular operator and we give an asymptotic formula for the negative moments of a 2 for any diagonal a we use a mixture of complex analytic and combinatorial techniques each giving finer information where the other can give only coarse detail in particular we introduce partition structure diagrams in section 4 a new combinatorial structure arising in free probability

uniqueness of solutions of the kdv hierarchy via dubrovin type flows abstract we consider the cauchy problem for the kdv hierarchy a family of integrable pdes with a lax pair representation involving one dimensional schrodinger operators under a local in time boundedness assumption on the solution for reflectionless initial data we prove that the solution stays reflectionless for almost periodic initial data with absolutely continuous spectrum we prove that under craig type conditions on the spectrum dirichlet data evolve according to a lipschitz dubrovin type flow so the solution is uniquely recovered by a trace formula this applies to algebro geometric finite gap solutions more notably we prove that it applies to small quasiperiodic initial data with analytic sampling functions and diophantine frequency this also gives a uniqueness result for the cauchy problem on the line for periodic initial data even in the absence of craig type conditions

a survey and some extensions of t chan s preconditioner abstract for any given n by n matrix an t chan s circulant preconditioner cf an proposed by t chan t chan an optimal circulant preconditioner for toeplitz systems siam j sci statist comput 9 1988 766 771 is defined to be the solution of min c n a n c n f over all n by n circulant matrices cn the cf an called the optimal circulant preconditioner by t chan has been proved to be a good preconditioner for a large class of structured systems a generalization of t chan s circulant preconditioner was given by huckle t huckle circulant and skew circulant matrices for solving toeplitz matrix problems siam j matrix anal appl 13 1992 767 777 which can be used for solving some general linear systems in this paper we review some old and develop some new properties of t chan s circulant preconditioner

floquet solutions for the 1 dimensional quasi periodic schrodinger equation we show that the 1 dimensional schrodinger equation with a quasiperiodic potential which is analytic on its hull admits a floquet representation for almost every energye in the upper part of the spectrum we prove that the upper part of the spectrum is purely absolutely continuous and that for a generic potential it is a cantor set we also show that for a small potential these results extend to the whole spectrum

superoptimal preconditioned conjugate gradient iteration for image deblurring we study the superoptimal frobenius operators in the two level circulant algebra we consider two specific viewpoints 1 the regularizing properties in imaging and 2 the computational effort in connection with the preconditioned conjugate gradient method some numerical experiments illustrating the effectiveness of the proposed technique are given and discussed

almost periodicity in time of solutions of the kdv equation we study the cauchy problem for the kdv equation tu 6u xu xu 0 with almost periodic initial data u x 0 v x we consider initial data v for which the associated schrodinger operator is absolutely continuous and has a spectrum that is not too thin in a sense we specify and show the existence uniqueness and almost periodicity in time of solutions this establishes a conjecture of percy deift for this class of initial data the result is shown to apply to all small analytic quasiperiodic initial data with diophantine frequency vector

r transforms of free joint distributions and non crossing partitions abstract we introduce an extension of the notion of r transform defined by d voiculescu to free joint distributions i e normalized linear functionals on algebras of non commutative polynomials in several indeterminates we point out that the r transform has good behavior with respect to the operations of free product and free convolution of joint distributions we prove that the explicit computation of the inverse r transform of a joint distribution is done via a formula of summation over the lattice of non crossing partitions which shows that the r transform is the operator theoretic counterpart of the free cumulants considered in the combinatorial approach of r speicher moreover in connection to the equivariance of the multidimensional r transform under rotations we show that a natural free analogue of a classical result about rotations of independent random variables is holding

generic singular spectrum for ergodic schrodinger operators we consider schrodinger operators with ergodic potential v n f t n n z where t is a nonperiodic homeomorphism we show that for generic f c the spectrum has no absolutely continuous component the proof is based on approximation by discontinuous potentials which can be treated via kotani theory

generalized reflection coefficients i consider general reflection coefficients for arbitrary one dimensional whole line differential or difference operators of order 2 these reflection coefficients are semicontinuous functions of the operator their absolute value can only go down when limits are taken this implies a corresponding semicontinuity result for the absolutely continuous spectrum which applies to a very large class of maps in particular we can consider shift maps thus recovering and generalizing a result of last simon and flows of the toda and kdv hierarchies this is new finally i evaluate an attempt at finding a similar general setup that gives the much stronger conclusion of reflectionless limit operators in more specialized situations

nonuniqueness for solutions of the korteweg de vries equation des variantes de la methode de diffusion inverse donnent des exemples de non unicite pour le probleme de cauchy pour l equation de korteweg de vries

some stability properties of t chan s preconditioner abstract a matrix is said to be stable if the real parts of all the eigenvalues are negative in this paper for any matrix a n we give some sufficient and necessary conditions for the stability of t chan s preconditioner c u a n

primitive potentials and bounded solutions of the kdv equation abstract we construct a broad class of bounded potentials of the one dimensional schrodinger operator that have the same spectral structure as periodic finite gap potentials but that are neither periodic nor quasi periodic such potentials which we call primitive are non uniquely parametrized by a pair of positive holder continuous functions defined on the allowed bands primitive potentials are constructed as solutions of a system of singular integral equations which can be efficiently solved numerically simulations show that these potentials can have a disordered structure primitive potentials generate a broad class of bounded non vanishing solutions of the kdv hierarchy and we interpret them as an example of integrable turbulence in the framework of the kdv equation

the isospectral torus of quasi periodic schrodinger operators via periodic approximations we study the quasi periodic schrodinger operator begin aligned psi x v x psi x e psi x quad x in mathbb r end aligned in the regime of small v x sum m in mathbb z nu c m exp 2 pi i m omega x omega omega 1 dots omega nu in mathbb r nu c m le varepsilon exp kappa 0 m we show that the set of reflectionless potentials isospectral with v is homeomorphic to a torus moreover we prove that any reflectionless potential q isospectral with v has the form q x sum m in mathbb z nu d m exp 2 pi i m omega x with the same omega and with d m le sqrt 2 varepsilon exp frac kappa 0 2 m our derivation relies on the study of the approximation via hill operators with potentials tilde v x sum m in mathbb z nu c m exp 2 pi i m tilde omega x where tilde omega is a rational approximation of omega it turns out that the multi scale analysis method of damanik and goldstein publ math inst hautes etudes sci 119 217 401 2014 applies to these hill operators namely in damanik et al trans am math soc to appear arxiv 1409 2147 2016 we developed the multi scale analysis for the operators dual to the hill operators in question the main estimates obtained in damanik et al trans am math soc to appear arxiv 1409 2147 2016 allow us here to establish estimates for the gap lengths and the fourier coefficients in a form that is considerably stronger than the estimates known in the theory of hill operators with analytic potentials in the general setting due to these estimates the approximation procedure for the quasi periodic potentials is effective despite the fact that the rate of approximation omega tilde omega thicksim tilde t delta 0 delta 1 2 is slow on the scale of the period tilde t of the hill operator

generic continuous spectrum for ergodic schrodinger operators we consider families of discrete schrodinger operators on the line with potentials generated by a homeomorphism on a compact metric space and a continuous sampling function we introduce the concepts of topological and metric repetition property assuming that the underlying dynamical system satisfies one of these repetition properties we show using gordon s lemma that for a generic continuous sampling function the set of elements in the associated family of schrodinger operators that have no eigenvalues is large in a topological or metric sense respectively we present a number of applications particularly to shifts and skew shifts on the torus

cumulants in noncommutative probability theory ii we continue the investigation of noncommutative cumulants in this paper various characterizations of generalized gaussian random variables are proved

global theory of one frequency schrodinger operators we study schrodinger operators with a one frequency analytic potential focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials from the dynamical point of view the transition signals the emergence of non uniform hyperbolicity so the dependence of the lyapunov exponent with respect to parameters plays a central role in the analysis though often ill behaved by conventional measures we show that the lyapunov exponent is in fact remarkably regular in a stratified sense which we define the irregularity comes from the matching of nice analytic or smooth functions along sets with complicated geometry this result allows us to establish that the critical set for the transition lies within countably many codimension one subvarieties of the infinite dimensional parameter space a more refined renormalization based analysis shows that the critical set is rather thin within those subvarieties and allows us to conclude that a typical potential has no critical energies such acritical potentials also form an open set and have several interesting properties only finitely many phase transitions may happen but never at any specific point in the spectrum and the lyapunov exponent is minorated in the region of the spectrum where it is positive on the other hand we do show that the number of phase transitions can be arbitrarily large

renormalized oscillation theory for hamiltonian systems abstract we extend a result on renormalized oscillation theory originally derived for sturm liouville and dirac type operators on arbitrary intervals in the context of scalar coefficients to the case of general hamiltonian systems with block matrix coefficients in particular this contains the cases of general sturm liouville and dirac type operators with block matrix valued coefficients as special cases the principal feature of these renormalized oscillation theory results consists in the fact that by replacing solutions by appropriate wronskians of solutions oscillation theory now applies to intervals in essential spectral gaps where traditional oscillation theory typically fails

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

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

sample variance in free probability abstract let x 1 x 2 x n denote i i d centered standard normal random variables then the law of the sample variance q n i 1 n x i x 2 is the 2 distribution with n 1 degrees of freedom it is an open problem in classical probability to characterize all distributions with this property and in particular whether it characterizes the normal law in this paper we present a solution of the free analogue of this question and show that the only distributions whose free sample variance is distributed according to a free 2 distribution are the semicircle law and more generally so called odd laws by which we mean laws with vanishing higher order even cumulants in the way of proof we derive an explicit formula for the free cumulants of q n which shows that indeed the odd cumulants do not contribute and which exhibits an interesting connection to the concept of r cyclicity

the inverse periodic problem for hill s equation with a finite gap potential let h q d d q x be the hill s operator with a periodic potential q of period one consider the following inverse problem find all potentials q x with the same periodic spectrum as a given potential p x two explicit solutions are known for finite gap potentials its and matveev 6 give an explicit formula in terms of theta functions their solution was generalized by mckean and trubowitz lo to infinite gap potentials finkel et al 5 give a second formula which expresses q in terms of the base potential p the dirichlet eigenfunctions of p and the floquet solutions of p the formula involves no algebraic geometry in this paper we represent finite gap potentials q as rational functions of the base potential p and the derivatives ofp the representation makes use of the finkel isaacson trubowitz fit formula

a new example of independence and white noise we examine the notion of free independence according to voiculescu this form of independence is used for defining free white noise or process with stationary and freely independent increments we prove a general limit theorem giving the combinatorics of infinitely freely divisible states and thus of free white noises with the help of admissible partitions we realize the free analogues of the wiener process and of the poisson process as processes on the full fock space ofl2