effective masses and conformal mappings letgn n n denote the set of gaps of the hill operator we solve the following problems 1 find the effective massesmn 2 compare the effective massmn with the length of the gapgn and with the height of the corresponding slit on the quasimomentum plane both with fixed numbern and their sums 3 consider the problems 1 2 for more general cases the dirac operator with periodic coefficients the schrodinger operator with a limit periodic potential to obtain 1 3 we use a conformal mapping corresponding to the quasimomentum of the hill operator or the dirac operator

mkdv and its birkhoff coordinates abstract we show that the modified korteweg de vries equation with periodic boundary conditions admits global birkhoff coordinates on any soblev space h n with n 1 the construction of these coordinates involves a new lax pair for mkdv

marchenko ostrovki mapping for periodic zakharov shabat systems abstract consider the zakharov shabat or dirac operator t in the hilbert space l2 r l2 r with real periodic vector potential q q1 q2 h l2 t l2 t the spectrum of t is absolutely continuous and consists of intervals separated by gaps gn z n z n n z from the dirichlet eigenvalue mn n z of the zakharov shabat equation with dirichlet boundary conditions at 0 1 the square of the height of vertical slits on the quasimomentum domain and the points on these slits we construct the marchenko ostrovki vertical slits mapping for the periodic zakharov shabat systems h h l2 l2 using nonlinear functional analysis in hilbert spaces we show that this mapping is a real analytic isomorphism in the second part of our paper we prove a new identity for the effective masses

spectral asymptotics for schrodinger operators with periodic point interactions abstract spectrum of the second order differential operator with periodic point interactions in l 2 r is investigated classes of unitary equivalent operators of this type are described spectral asymptotics for the whole family of periodic operators are calculated it is proven that the first several terms in the asymptotics determine the class of equivalent operators uniquely it is proven that the spectrum of the operators with anomalous spectral asymptotics when the ratio between the lengths of the bands and gaps tends to zero at infinity can be approximated by standard periodic weighted operators with step wise density functions it is shown that this sequence of periodic weighted operators converges in the norm resolvent sense to the formal generalized resolvent of the periodic schrodinger operator with certain energy dependent boundary conditions the operator acting in an extended hilbert space such that its resolvent restricted to l 2 r coincides with the formal resolvent is constructed explicitly

positive lyapunov exponents for schrodinger operators with quasi periodic potentials we present a new simple way to estimate the rate of exponential growth lyapunov exponent of solutions of the finite difference schrodinger equation h e psi n mathop limits def psi n 1 psi n 1 lambda f alpha n theta psi n heref is a non constant real analytic function of period 1 and is irrational for large we prove that the lyapunov exponent is positive for every energye in the spectrum ofh and a e in particular the absolutely continuous spectrum ofh is empty in the continuum we study the quasi periodic operator onl2 r h frac d 2 dx 2 k 2 cos x cos alpha x theta for largek and show that for wide intervals of low energies the lyapunov exponent is positive the main idea which originated from m herman s subharmonic argument 11 is to deform the phase to the complex plane this enables us to avoid small denominator problems by moving them off the axis making estimates much easier to perform we recover the information for real using an elementary extension of jensen s formula subharmonicity

the estimates of periodic potentials in terms of effective masses letgn an an n 1 denote the gaps mn be the effective masses and n an 1 an a0 0 be the spectral bands of the hill operatort d2 dx2 v x inl2 r wherev is a 1 periodic real potential froml2 0 1 let the length gapln gn hn be the height of the corresponding slit on the quasimomentum domain and n 2 2n 1 n 0 be the band reduction let l n sqrt a n sqrt a n n geqq 1 n 1 denote the gap length for the operator sqrt t geqq 0 introduce the sequencesl ln h hn l ln n m mn and the norms left f right m 2 sum nolimits n 0 left 2 pi n right 2m f n 2 m geqq 0 m 0 the following results are obtained i the estimates of v l h 1 l 1 in terms of m 2 ii identities for the dirichlet integral of quasimomentum and integral of potentials and so on iii the generation of i ii for more general potentials

on the deficiency indices and self adjointness of symmetric hamiltonian systems abstract the main purpose of this paper is to investigate the formal deficiency indices n of a symmetric first order system jf bf h f on an interval i where i r or i r here j b h are n n matrix valued functions and the hamiltonian h 0 may be singular even everywhere we obtain two results for such a system to have minimal numbers n 0 if i r resp n n if i r and a criterion for their maximality n 2n for i r as well as the quasi regularity this covers the kac krein and de branges trans amer math soc 99 1961 118 theorems on 2 2 canonical systems and some results from kogan and rofe beketov proc roy soc edinburgh sect a 74 1974 75 5 some conditions for a canonical system to have intermediate formal deficiency indices are presented too we also obtain a generalization of the well known titchmarsh sears theorem for second order sturm liouville type equations this contains results due to lidskii and krein as special cases we present two approaches to the above problems one dealing with formal deficiency indices and one dealing with ordinary deficiency indices our main non formal approach is based on the investigation of a symmetric linear relation smin which is naturally associated to a first order system this approach works in the framework of extension theory and therefore we investigate in detail the domain d s min of s min in particular we prove the so called regularity theorem for d s min as a byproduct of the regularity result we obtain very short proofs of generalizations of the main results of the paper by kogan and rofe beketov 1974 75

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

a note on a lower bound for the smallest singular value abstract we give a short proof of a lower bound for the smallest singular value mentioned by hong and pan one of our results improves their theorem we obtain another lower bound that is sharp an example shows that these results can be quite accurate

resonances for steplike potentials forward and inverse results we consider resonances associated to the one dimensional schrodinger operator d 2 d x 2 v x where v x v if x x m and v x v if x x m with v v we obtain asymptotics of the resonance counting function for several regions moreover we show that in several situations the resonances v and v determine v uniquely up to translation

estimates of periodic potentials in terms of gap lengths define the hill operator t d 2 dx 2 q x in l 2 and suppose q l 2 0 1 is a 1 periodic real potential 0 1 q x dx 0 we prove the estimate 2 1 1 3 where 2 n 1 n 2 and n 0 n 1 is the gap length of t

a generalized langevin equation for dealing with nonadditive fluctuations a suitable extension of the mori memory function formalism to the non hermitian case allows a multiplicative process to be described by a langevin equation of non markoffian nature this generalized langevin equation is then shown to provide for the variable of interest the same autocorrelation function as the well known theoretical approach developed by kubo the stochastic liouville equation sle theory it is shown furthermore that the present approach does not disregard the influence of the variable of interest on the time evolution of its thermal bath the stochastic process under study can also be described by a fokker planck like equation which results in a gaussian equilibrium distribution for the variable of interest the main flaw of the sle theory that resulting in an uncorrect equilibrium distribution is therefore completely eliminated

on the absolutely continuous spectrum of one dimensional quasi periodic schrodinger operators in the adiabatic limit cet article est consacre a l etude du spectre de certaines familles d equations de schrodinger quasi periodiques sur l axe reel lorsque les varietes iso energetiques adiabatiques sont etendues dans la direction des positions nous demontrons que dans un intervalle d energie ou ceci est le cas le spectre est dans sa majeure partie purement absolument continu et que les fonctions propres generalisees correspondantes sont des fonctions de bloch floquet

non normal type singular integral differential equations by riemann hilbert approach abstract this article deals with one class of singular integral differential equations of non normal type with convolution and cauchy principal value integral in class 0 by using fourier transform this classes of equations are transformed into a riemann boundary value problem with nodal point and we prove the existence of solutions and the noether theory for the equations for such equations we propose one method different from classical one and we obtain the general solutions and the conditions of solvability all cases about the behaviors of the solution are considered at nodal points therefore the theory of integral equations and the classical riemann boundary value problems is extended further

periodic weighted operators abstract consider the periodic weighted operator ty 2 2 y in l 2 r x 2 dx where the real function is 1 periodic positive and let q l 2 0 1 the spectrum of t consists of intervals separated by gaps n n 1 with the lengths n let h n be a height of the corresponding slit in the quasimomentum domain and let g n n 1 be the gap with the length g n of the operator t 0 the following results are obtained i the quasimomentum k for the weighted operator t is constructed and the basic properties of k are studied ii two sided estimates of l 2 norms of the sequences n n 1 h n 1 g n 1 in terms of 0 1 q x 2 dx iii the asymptotics of the gap length n as n are determined the proofs are based on the analysis of the quasimomentum as a conformal mapping embedding theorems and the identities between the quasimomentum and the potential in order to prove these results the asymptotics of the fundamental solutions and the lyapunov function are obtained at high energy

effective masses for laplacians on periodic graphs abstract we consider laplacians on both periodic discrete and periodic metric equilateral graphs their spectrum consists of an absolutely continuous part which is a union of non degenerate spectral bands and flat bands i e eigenvalues of infinite multiplicity we estimate effective masses associated with the ends of each spectral band in terms of geometric parameters of the graphs moreover in the case of the bottom of the spectrum we determine two sided estimates on the effective mass in terms of geometric parameters of the graphs the proof is based on floquet theory factorization of fiber operators perturbation theory and the relation between effective masses for laplacians on discrete and metric graphs obtained in our paper

on the oscillation of hill s equations under periodic forcing abstract we use the floquet theory of the hill s equation to prove the conjecture that all solutions of the second order forced linear differential equation y c sin t y cos t are oscillatory on 0 for all c 0

positivity preserving non markovian master equations a general class of integrodifferential non markovian master equations is developed which is representative of the dynamics of small subsystems interacting with open reservoirs with memory conditions which guarantee positivity of the subsystem reduced density are established

the solvability and explicit solutions of singular integral differential equations of non normal type via riemann hilbert problem abstract in this article we study non normal type singular integral differential equations involving convolution kernel and cauchy kernel in class 0 and we prove the existence of solutions and the noether conditions by using theory of fourier analysis the equations under consideration are transformed into boundary value problems for analytic functions i e riemann hilbert problems with discontinuous coefficients for such problems we propose one method different from classical one and we obtain the analytic solutions and the conditions of noether solvability at nodal points all cases as regards the regularity and the asymptotic behaviors of the solution are discussed in detail in particular we generalize the case of normal type discussed in the literature li 2017 7 8 2020 2019 to the case of non normal type

asymptotic solution for first and second order linear volterra integro differential equations with convolution kernels this paper addresses the problem of finding an asymptotic solution for first and second order integro differential equations containing an arbitrary kernel by evaluating the corresponding inverse laplace and fourier transforms the aim of the paper is to go beyond the tauberian theorem in the case of integral differential equations which are widely used by the scientific community the results are applied to the convolute form of the lindblad equation setting generic conditions on the kernel in such a way as to generate a positive definite density matrix and show that the structure of the eigenvalues of the correspondent liouvillian operator plays a crucial role in determining the positivity of the density 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

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

distribution of poles for scattering on the real line abstract the density of scattering poles is shown to be proportional to the length of the convex hull of the support of the potential in the case of a potential with finite singularities at the endpoints of the support asymptotic formulae for the poles are given while in the c 0 case an example of a potential with infinitely many scattering poles on i r is constructed the scattering amplitude of a compactly supported potential is also characterized

the discrete spectrum in the gaps of the continuous one for non signdefinite perturbations with a large coupling constant given two selfadjoint operators a and v v v we study the motion of the eigenvalues of the operator a t a tv as t increases let 0 and let be a regular point for a we consider the quantities n n n 0 defined as the number of the eigenvalues of the operator a t that pass point from the right to the left from the left to the right or change the direction of their motion exactly at point respectively as t increases from 0 to 0 an abstract theorem on the asymptotics for these quantities is presented applications to schrodinger operators and its generalizations are given

lattice dislocations in a 1 dimensional model the spectral properties of the schrodinger operator t t d2 dx2 q x t in l2 are studied where the potential q is defined by q p x t x 0 and q p x x 0 p is a 1 periodic potential and t is the dislocation parameter for each t the absolutely continuous spectrum ac t t ac t 0 consists of intervals which are separated by the gaps n t t n t 0 n n n 1 we prove in each gap n n 1 there exist two unique states an eigenvalue and a resonance n t of the dislocation operator such that n 0 n and the point n t runs clockwise around the gap n changing the energy sheet whenever it hits n making n 2 complete revolutions in unit time on the first sheet n t is an eigenvalue and on the second sheet n t is a resonance in general these motions are not monotonic there exists a unique state 0 t in the basic gap 0 t t 0 t 0 0 the asymptotics of n t as n is determined

nonlinear heat conduction through an externally heated radiant plasma background analysis for a numerical study a family of nonlinear parabolic problems of heat conduction through a radiant plasma is studied as in the technological application the governing equation contains a time dependent source term then considering boundary value problems the boundary conditions are time dependent too certain qualitative properties of solutions to such problems have been established namely number of local maxima lie symmetries and similarity solutions

generalized convolution type singular integral equations in this paper we study one class of generalized convolution type singular integral equations in class 0 such equations are turned into complete singular integral equations with nodal points and further turned into boundary value problems for analytic function with discontinuous coefficients by fourier transforms for such equations we will propose one method different from classical one and obtain the general solutions and their conditions of solvability in class 0 thus this paper generalizes the theory of classical equations of convolution type

invariants for laplacians on periodic graphs we consider a laplacian on periodic discrete graphs its spectrum consists of a finite number of bands in a class of periodic 1 forms i e functions defined on edges of the periodic graph we introduce a subclass of minimal forms with a minimal number mathcal i of edges in their supports on the period we obtain a specific decomposition of the laplacian into a direct integral in terms of minimal forms where fiber laplacians matrices have the minimal number 2 mathcal i of coefficients depending on the quasimomentum and show that the number mathcal i is an invariant of the periodic graph using this decomposition we estimate the position of each band the lebesgue measure of the laplacian spectrum and the effective masses at the bottom of the spectrum in terms of the invariant mathcal i and the minimal forms in addition we consider an inverse problem we determine necessary and sufficient conditions for matrices depending on the quasimomentum on a finite graph to be fiber laplacians moreover similar results for schrodinger operators with periodic potentials are obtained

the inverse problem for the hill operator a direct approach we consider the hill operator t yd2 dx2 v x on l2 r where v 2 l2 0 1 is an even 1 periodic real potential and r 1 0 v x dx 0 it is wellknown that the spectrum of t is absolutely continuous and consists of intervals s1 s2 such that sn any1 an any1 an an n 1 these intervals are separated by gaps g1 g2 where gn an an if a gap degenerates that is gn then the corresponding segments sn sn 1 merge let u x e v x e v be the solutions of the equation yf 00 vf ef e 2 c 1 1

solvability of singular integro differential equations via riemann hilbert problem abstract the article is to study singular integro differential equations involving convolutional operators and cauchy integral operators via riemann hilbert problem to do this we adopt a new approach through fourier transform on l 2 subspace which is holder continuous with a certain decay at infinity the fourier transform converts the equations into a riemann hilbert problem with holder continuous coefficients and with nodal points which allows us to construct the general solutions

inverse problem and the trace formula for the hill operator ii consider the hill operator t d 2 dx 2 q x on l 2 mathbb r where q in l 2 0 1 is a 1 periodic real potential and int 0 1q x dx 0 the spectrum of t is absolutely continuous and consists of intervals separated by gaps gamma n a n a n n geq 1 let mu n n geq 1 be the dirichlet eigenvalue of the equation y qy ly on the interval 0 1 introduce the vector g n g cn g sn in mathbb r 2 with components g cn a n a n 2 mu n and g sn gamma n 2 4 g cn 2 s n where the sign s n or s n for all n geq 1 using nonlinear functional analysis in hilbert spaces we show that the mapping g q to g q g n 1 iy in ell 2 oplus ell 2 is a real analytic isomorphism in the second part a trace formula for q in l 2 0 1 is proved

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

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

estimates for the hill operator ii consider the hill operator ty y q t y in l2 r where the real potential q is 1 periodic and q q l2 0 1 the spectrum of t consists of spectral bands separated by gaps n n 1 with length n 0 we obtain two sided estimates of the gap lengths n2 n 2 in terms of 01q t 2dt moreover we obtain the similar two sided estimates for spectral data the height of the corresponding slit on the quasimomentum domain action variables for the kdv equation and so on in order prove this result we use the analysis of a conformal mapping corresponding to quasimomentum of the hill operator that makes it possible to reformulate the problems for the differential operator as the problems of the conformal mapping theory then the proof is based on the analysis of the conformal mapping the embedding theorems and the identities furthermore we obtain the similar two sided estimates for potentials which have p 2 derivatives

spectral shift function and resonances for slowly varying perturbations of periodic schrodinger operators we study the spectral shift function s h and the resonances of the operator p h v x w hx here v is a periodic potential w a decreasing perturbation and h a small positive constant we give a representation of the derivative of s h related to the resonances of p h and we obtain a weyl type asymptotics of s h we establish an upper bound o h n 1 for the number of the resonances of p h lying in a disk of radius h

metal insulator transition for the almost mathieu operator we prove that for diophantine and almost every the almost mathieu operator h n n 1 ni 1 cos 2 n n exhibits localization for 2 and purely absolutely continuous spectrum for 2

homogenization of the schrodinger equation and effective mass theorems we study the homogenization of a schrodinger equation with a large periodic potential denoting by the period the potential is scaled as 2 we obtain a rigorous derivation of so called effective mass theorems in solid state physics more precisely for well prepared initial data concentrating on a bloch eigenfunction we prove that the solution is approximately the product of a fast oscillating bloch eigenfunction and of a slowly varying solution of an homogenized schrodinger equation the homogenized coefficients depend on the chosen bloch eigenvalue and the homogenized solution may experience a large drift the homogenized limit may be a system of equations having dimension equal to the multiplicity of the bloch eigenvalue our method is based on a combination of classical homogenization techniques two scale convergence and suitable oscillating test functions and of bloch waves decomposition

instability zones of a periodic 1d dirac operator and smoothness of its potential let l be the differential operator where p x q x are 1 periodic functions such that the operator l considered on 0 1 with periodic y 0 y 1 or antiperiodic y 0 y 1 boundary conditions is self adjoint and moreover for large n it has close to n a pair of periodic if n is even or antiperiodic if n is odd eigenvalues n n we study the relationship between the decay rate of the instability zone sequence n n n n and the smoothness of the potential function p x

nonlinear waves in shallow honeycomb lattices the linear spectrum and corresponding bloch modes of shallow honeycomb lattices near dirac points are investigated via perturbation theory the dispersion relation is found to have threefold degeneracy at leading order with eigenvalue splitting at the following two orders i e the threefold eigenvalue splits into single and double values multiscale perturbation methods are employed to describe the nonlinear dynamics of the associated wave envelopes the dynamics of the envelope depends on different asymptotic balances whereupon a three level nonlinear dirac type equation or a two level nonlinear dirac equation is derived the analysis agrees well with direct numerical simulations

on the resonances and eigenvalues for a 1d half crystal with localised impurity we consider the schrdinger operator h on the half line with a periodic potential p plus a compactly supported potential q for generic p its essential spectrum has an infinite sequence of open gaps we determine the asymptotics of the resonance counting function and show that for sufficiently high energy each non degenerate gap contains exactly one eigenvalue or antibound state giving asymptotics for their positions conversely for any potential q and for any sequences n 0 1 and xn 0 there exists a potential p such that xn is the length of the n th gap n and h has exactly n eigenvalues and 1 n antibound state in each high energy gap moreover we show that between any two eigenvalues in a gap there is an odd number of antibound states and hence deduce an asymptotic lower bound on the number of antibound states in an adiabatic limit

reduced dynamics need not be completely positive the reduced dynamics of a quantum system in contact with a reservoir is generally thought to be completely positive this is certainly true if product initial conditions are used to define the dynamics we show that with correlated initial conditions it need not be so in this case the dynamics can properly be defined only on a subset of initial system states extension by linearity to all possible initial states is trivially possible but the extension may not be physically realizable and may not even be positive let alone completely positive

estimates for periodic zakharov shabat operators abstract we consider the periodic zakharov shabat operators on the real line the spectrum of this operator consists of intervals separated by gaps with the lengths g n 0 n z let n be the corresponding effective masses and let h n be heights of the corresponding slits in the quasi momentum domain we obtain a priori estimates of sequences g g n n z n n z h h n n z in terms of weighted l p norms at p 1 the proof is based on the analysis of the quasi momentum as the conformal mapping

riccati map on l02 t and its applications abstract this paper is concerned with the spectral properties of the schrodinger operator l q def d 2 d x 2 q with periodic potential q from the sobolev space h 1 t we obtain asymptotic formulas and a priori estimates for the periodic and dirichlet eigenvalues which generalize known results for the case of potentials q l 2 t the key idea is to reduce the problem to a known one the spectrum of the impedance operator via a nonlinear analytic isomorphism between l 0 2 t and the sobolev space h 0 1 t

group classification and symmetry reduction of variable coefficient nonlinear diffusion convection equation based on classical lie group method the group for a general class of variable coefficient nonlinear diffusion convection equation in 1 1 dimensions is obtained new symmetries are found seven models have been studied

bounds on scattering poles in one dimension for the class of super exponentially decaying potentials on the real line sharp upper bounds on the counting function of the poles in discs are derived and the density of the poles in strips is estimated in the case of nonnegative potentials explicit estimates for the width of a pole free strip are obtained

inverse problem for periodic weighted operators define the periodic weighted operator ty 2 2y in l2 r x 2 dx suppose a function w21 r z is 1 periodic real positive 0 1 and let q l2 0 1 the spectrum of t consists of intervals n n 1 n separated by gaps n n n n 1 with the lengths n and 0 0 let m2n n 1 be the dirichlet eigenvalue of the equation y 2qy z2y y 0 y 1 0 where mn 0 introduce the lyapunov function z q for t and note that z zn q 0 for some zn n n let x z q be the solution of the equation 2q z2 z c satisfying 0 z q 0 0 z q 1 introduce the vector hn hcn hsn r2 with components hcn log 1 n 1 mn q hsn hn 2 hcn 2 1 2 sign zn mn where hn is defined by the equation cosh hn 1 n zn q 1 and coincides with the euclidien norm of the vector hn using nonlinear functional analysis in hilbert space we prove that the mapping h q h q hn 1 is a real analytic isomorphism

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

decoherence from internal degrees of freedom for clusters of mesoparticles a hierarchy of master equations a mesoscopic evolution equation for an ensemble of mesoparticles follows from the elimination of internal degrees of freedom if the system is composed of a hierarchy of scales then the reduction procedure could be worked repeatedly and the characterization of this iterating method is carried out namely a prescription describing a discrete hierarchy of master equations for the density operator is obtained decoherence follows from the irreversible coupling between the system defined by mesoscopic variables and internal degrees of freedom we briefly discuss the existence of systems with the same dynamics laws at different scales we made an explicit calculation for an ensemble of particles with internal harmonic interaction in an external anharmonic field new conditions related to the semiclassical limit for mesoscopic systems wigner function are conjectured

inverse spectral theory for uniformly open gaps in a weighted sturm liouville problem abstract motivated by a pde existence problem we study the inverse problem for a weighted sturm liouville operator l s associated with the eigenvalue problem y s x y 0 where s is a real valued periodic even function that is bounded from below by a positive constant and belongs to the l 2 based sobolev space h r 0 1 r 1 choosing gap lengths and gap midpoints as coordinates we define a spectral map g that assigns to a coefficient s the structure of the spectrum of l s we find that g is a real analytic isomorphism locally around s 1 which in particular implies the existence of coefficients s h r 0 1 r 3 2 whose spectrum features band structure with all gaps uniformly open around the gap midpoints this result paves the way for the construction of so called breathers in nonlinear wave equations with such coefficients s apart from the novelty of treating the inverse spectral problem for the full banach scale h r 0 1 r 1 the local nature of our result allows more concise and transparent proofs in particular instead of using any preliminary transformations we treat the weighted problem directly by adapting techniques used for schrodinger operators with distribution potentials

anderson transitions for a family of almost periodic schrodinger equations in the adiabatic case this work is devoted to the study of a family of almost periodic one dimensional schrodinger equations using results on the asymptotic behavior of a corresponding monodromy matrix in the adiabatic limit we prove the existence of an asymptotically sharp anderson transition in the low energy region more explicitly we prove the existence of energy intervals containing only singular spectrum and of other energy intervals containing absolutely continuous spectrum the zones containing singular spectrum and those containing absolutely continuous are separated by asymptotically sharp transitions the analysis may be viewed as utilizing a complex wkb method for adiabatic perturbations of periodic schrodinger equations the transition energies are interpreted in terms of phase space tunneling

reducibility or nonuniform hyperbolicity for quasiperiodic schrodinger cocycles we show that for almost every frequency r q for every c o potential v r z r and for almost every energy e the corresponding quasiperiodic schr nodinger cocycle is either reducible or nonuniformly hyperbolic this result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic schr nodinger operator and allows us to complete the proof of the aubry andr le conjecture on the measure of the spectrum of the almost mathieu operator

resonances for dirac operators on the half line abstract we consider the 1d dirac operator on the half line with compactly supported potentials we study resonances as the poles of scattering matrix or equivalently as the zeros of modified fredholm determinant we obtain the following properties of the resonances 1 asymptotics of counting function 2 estimates on the resonances and the forbidden domain

solvability theory of convolution singular integral equations via riemann hilbert approach abstract in this paper we study the solvability for some classes of singular integral equations of convolution type with discontinuous property in class 0 though such equations can be solved by using the classical bekya regularization method we are to give a new sectionally jumping method that is they may be reduced to boundary value problems of holomorphic functions which can be systematically solved and easily discussed we obtain the general solutions and the conditions of their solvability thus this paper generalizes the classical theory of integral equations

tunnelling between tori in phase space abstract this paper calculates the energy splitting e due to tunnelling between a pair of quantum states which correspond to classical motion on tori in phase space the calculation assumes that the states are separated in coordinate space however the result of the calculation e a 3 2 e 2 h has canonically invariant expressions for the tunnelling actions s and prefactor a and it is conjectured that the results apply even when the tori overlap in coordinate space this makes the results relevant to a very general problem connected with the einstein brillouin keller quantization scheme there is a surprising consequence of this calculation when the system is separable it is clear that e vanishes for most pairs of states this does not occur because the prefactor a vanishes but because the tunnelling action s is infinite as a parameter is varied to make the system separable s diverges logarithmically the results of this paper only apply to quasi integrable systems where s is finite p in the case of any exactly integrable system s is infinite and e vanishes in a more singular manner than e const h as 0

the marchenko ostrovski mapping and the trace formula for the camassa holm equation we consider the periodic weighted operator ty 2 2y 14 4 in l2 r 2dx where is a 1 periodic positive function satisfying q l2 0 1 the spectrum of t consists of intervals separated by gaps in the first part of the paper we construct the marchenko ostrovski mapping q h q and solve the corresponding inverse problem for our approach it is essential that the mapping h has the factorization h q h0 v q where q v q is a certain nonlinear mapping and v h0 v is the marchenko ostrovski mapping for the hill operator moreover we solve the inverse problem for the gap length mapping in the second part of this paper we derive the trace formula for t

asymptotic distribution of resonances in one dimension abstract we determine the leading asymptotics of the resonance counting function for a class of schrodinger operators in one dimension whose potentials may have non compact support

resonances in one dimension and fredholm determinants we discuss resonances for schrodinger operators in whole and half line problems one of our goals is to connect the fredholm determinant approach of froese to the fourier transform approach of zworski another is to prove a result on the number of antibound states namely in a half line problem there are an odd number of antibound states between any two bound states

conformal spectral theory for the monodromy matrix for any n x n monodromy matrix we define the lyapunov function which is analytic on an associated n sheeted riemann surface on each sheet the lyapunov function has the standard properties of the lyapunov function for the hill operator the lyapunov function has real or complex branch points which we call resonances we determine the asymptotics of the periodic anti periodic spectrum and of the resonances at high energy we show that the endpoints of each gap are periodic anti periodic eigenvalues or resonances real branch points moreover the following results are obtained 1 we define the quasimomentum as an analytic function on the riemann surface of the lyapunov function various properties and estimates of the quasimomentum are obtained 2 we construct the conformal mapping with imaginary part given by the lyapunov exponent and we obtain various properties of this conformal mapping which are similar to the case of the hill operator 3 we determine various new trace formulae for potentials and the lyapunov exponent 4 we obtain a priori estimates of gap lengths in terms of the dirichlet integral we apply these results to the schrodinger operators and to first order periodic systems on the real line with a matrix valued complex self adjoint periodic potential

nonlinear wave packets in deformed honeycomb lattices the spectrum of a schrodinger operator with a perfect honeycomb lattice potential has special points called dirac points where the lowest two branches of the spectrum touch deformations can result in the merging and disappearance of the dirac points and the originally intersecting dispersion relation branches separate corresponding to these deformations nonlinear envelope equations are derived and their dynamics are studied in the region where dirac points exist a maximally balanced equation is derived which has limits to a nonlinear schrodinger kadomtsev petviashvili nlskp type equation and its dispersionless reduction when the dirac points disappear and a gap opens a different maximally balanced equation is derived which has the nlskp equation and a one dimensional nonlocal evolution equation as limits when the gap is sufficiently wide a nonlinear dirac equation with nonzero mass and a nonlinear schrodinger focusing defocusing system are found the latter two equations admit nonlinear lo

asymptotics of the number of zeros and of the eigenvalues of general weighted sturm liouville problems soit n le nombre de zeros d une solution de l equation de sturm liouville ponderee generale p x y r x q x y 0 sur 0 b ou 0 b p q r 0 b r et 1 p q r l 0 b on donne une demonstration directe de l estimation asymptotique de n