holder continuity of the spectral measures for one dimensional schrodinger operator in exponential regime avila and jitomirskaya prove that the spectral measure v xf of quasi periodic schrodinger operator is 1 2 holder continuous with appropriate initial vector f if satisfies diophantine condition and is small in the present paper the conclusion is extended to that for all with the spectral measure v xf is 1 2 holder continuous with small if v is real analytic in a neighbor of x c where c is a large absolute constant in particular the spectral measure xf of almost mathieu operator is 1 2 holder continuous if e c with c a large absolute constant

isotropie dʼune forme bilineaire dʼalbert sur le corps de fonctions dʼune quadrique en caracteristique 2 abstract in this paper we give a complete answer to the isotropy of an albert bilinear form over the function field of a quadric in characteristic 2 as a consequence we complete the classification of nongood bilinear forms of height and degree 2 given in laghribi and rehmann 2009 8

bound states for nano tubes with a dislocation abstract as a model for an interface in solid state physics we consider two real valued potentials v 1 and v 2 on the cylinder or tube s r r z where we assume that there exists an interval a 0 b 0 which is free of spectrum of v k for k 1 2 we are then interested in the spectrum of h t v t for t r where v t x y v 1 x y for x 0 and v t x y v 2 x t y for x 0 while the essential spectrum of h t is independent of t we show that discrete spectrum related to the interface at x 0 is created in the interval a 0 b 0 at suitable values of the parameter t provided v 2 has some essential spectrum in a 0 we do not require v 1 or v 2 to be periodic we furthermore show that the discrete eigenvalues of h t are lipschitz continuous functions of t if the potential v 2 is locally of bounded variation

on the blow up rate and the blow up set of breaking waves for a shallow water equation we consider the problem of the development of singularities for classical solutions to a new periodic shallow water equation blow up can occur only in the form of wave breaking i e the solution remains bounded but its slope becomes unbounded in finite time a quite detailed description of the wave breaking phenomenon is given there is at least a point in general depending on time where the slope becomes infinite exactly at breaking time the precise blow up rate is established and for a large class of initial data we also determine the blow up set

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

localization for a class of one dimensional quasi periodic schrodinger operators we prove for small and satisfying a certain diophantine condition the operator h varepsilon 2 delta frac 1 2 pi cos 2 pi j alpha theta j in mathbb z has pure point spectrum for almost all a similar result is established at low energy for h frac d 2 dx 2 k 2 cos 2 pi x cos 2 pi alpha x theta providedk is sufficiently large

the spectrum of skew shift schrodinger operators contains intervals abstract i prove that the spectrum of a skew shift schrodinger operator contains large intervals in the semi classical regime in the semi classical limit these intervals approach the range of the potential

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

dynamics of the quasi periodic schrodinger cocycle at the lowest energy in the spectrum in this paper we consider the quasi periodic schrodinger cocycle over mathbb t d d 1 and in particular its projectivization in the regime of large coupling constants and diophantine frequencies we give an affirmative answer to a question posed by m herman 21 p 482 concerning the geometric structure of certain strange non chaotic attractors which appear in the projective dynamical system we also show that for some phase the lowest energy in the spectrum of the associated schrodinger operator is an eigenvalue with an exponentially decaying eigenfunction this generalizes 39 to the multi frequency case d 1

global weak solutions for a new periodic integrable equation with peakon solutions abstract we prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions

cantor spectrum for the almost mathieu equation for a dense g of pairs in r2 we prove that the operator hu n u n 1 u n 1 cos 2 n u n has a nowhere dense spectrum along the way we prove several interesting results about the case pq of which we mention a if q is not an integral multiple of then all gaps are open and b if q is even and is chosen suitably then the middle gap is closed for all

gauss polynomials and the rotation algebra newton s binomial theorem is extended to an interesting noncommutative setting as follows if in a ring ba ab with commuting witha andb then the generalized binomial coefficient left begin array 20 c n k end array right r arising in the expansion left a b right n sum limits k 0 n left begin array 20 c n k end array right gamma a n k b k resulting from these relations is equal to the value at of the gaussian polynomial left begin array 20 c n k end array right frac left n right left k right left n k right where m 1 x m 1 x m 1 1 x this is of course known in the case 1 from this it is deduced that in the universal c algebraa gq generated by unitariesu andv such thatvu e 2 i uv the spectrum of the self adjoint element u v u v has all the gaps that have been predicted to exist provided that either is rational or is a liouville number in the latter case the gaps are labelled in the natural way viak theory by the set of all non zero integers and the spectrum is a cantor set

holder continuity of the integrated density of states for quasi periodic schrodinger equations and averages of shifts of subharmonic functions in this paper we consider various regularity results for discrete quasiperiodic schr6dinger equations n l pn 1 v 9 nw on eon with analytic potential v we prove that on intervals of positivity for the lyapunov exponent the integrated density of states is holder continuous in the energy provided w has a typical continued fraction expansion the proof is based on certain sharp large deviation theorems for the norms of the monodromy matrices and the avalanche principle the latter refers to a mechanism that allows us to write the norm of a monodromy matrix as the product of the norms of many short blocks in the multi frequency case the integrated density of states is shown to have a modulus of continuity of the form exp log tl for some 0 a 1 but currently we do not obtain holder continuity in the case of more than one frequency we also present a mechanism for proving the positivity of the lyapunov exponent for large disorders for a general class of equations the only requirement for this approach is some weak form of a large deviation theorem for the lyapunov exponents in particular we obtain an independent proof of the herman sorets spencer theorem in the multi frequency case the approach in this paper is related to the recent nonperturbative proof of anderson localization in the quasi periodic case by j bourgain and m goldstein

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

cantor spectrum for the almost mathieu operator in this paper we use results on reducibility localization and duality for the almost mathieu operator left h b phi x right n x n 1 x n 1 b cos left 2 pi n omega phi right x n on l 2 and its associated eigenvalue equation to deduce that for b 0 2 and diophantine the spectrum of the operator is a cantor subset of the real line this solves the so called ten martini problem for these values of b and moreover we prove that for b 0 small or large enough all spectral gaps predicted by the gap labelling theorem are open

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

continuity of the lyapunov exponent for quasiperiodic operators with analytic potential we study regularity properties of the lyapunov exponent l of one frequency quasiperiodic operators with analytic potential under no assumptions on the diophantine class of the frequency we prove joint continuity of l in frequency and energy at every irrational frequency

genericity of mode locking for quasiperiodically forced circle maps abstract we show that a generic quasiperiodically forced circle homeomorphism is mode locked the rotation number in the fibres is rationally related to the rotation number in the base and it is stable under small perturbations of the system as a consequence this implies that for a generic parameter family of quasiperiodically forced circle homeomorphisms satisfying a twist condition the graph of the rotation number as a function of the parameter is a devil s staircase

on the spectrum of multi frequency quasiperiodic schrodinger operators with large coupling we study multi frequency quasiperiodic schrodinger operators on mathbb z we prove that for a large real analytic potential satisfying certain restrictions the spectrum consists of a single interval the result is a consequence of a criterion for the spectrum to contain an interval at a given location that we establish non perturbatively in the regime of positive lyapunov exponent

on optimal separation of eigenvalues for a quasiperiodic jacobi matrix we consider quasiperiodic jacobi matrices of size n with analytic coefficients we show that in the positive lyapunov exponent regime after removing some small sets of energies and frequencies any eigenvalue is separated from the rest of the spectrum by n 1 log n p with p 15