quantum singular complete integrability abstract we consider some perturbations of a family of pairwise commuting linear quantum hamiltonians on the torus with possibly dense pure point spectra we prove that their rayleigh schrodinger perturbation series converge near each unperturbed eigenvalue under the form of a convergent quantum birkhoff normal form moreover the family is jointly diagonalized by a common unitary operator explicitly constructed by a newton type algorithm this leads to the fact that the spectra of the family remain pure point the results are uniform in the planck constant near 0 the unperturbed frequencies satisfy a small divisors condition and we explicitly estimate how this condition can be released when the family tends to the unperturbed one in the case where the number of operators is equal to the number of degrees of freedom i e full integrability our construction provides convergent normal forms for general perturbations of linear systems

on the reducibility of linear quasi periodic systems with liouvillean basic frequencies and multiple eigenvalues abstract in this paper we consider the linear quasi periodic system x a p t x x r d where a is a d d constant matrix of elliptic type and has multiple eigenvalues p t is analytic quasi periodic with respect to t with basic frequencies 1 where is irrational and is a small perturbation parameter under suitable non resonant condition non degeneracy condition and 0 r where lim sup n ln q n 1 q n q n is the sequence of denominations of the best rational approximations for r q 0 r r r is the initial radius of analytic strip it is proved that for most sufficiently small this system can be reduced to a constant system x a x where a is a constant matrix close to a as some applications we apply our results to quasi periodic hill s equations three dimensional skew symmetric systems and n coupled schrodinger equations to study the lyapunov stability of the equilibrium and the existence of quasi periodic solutions

semiclassical analysis of low lying eigenvalues ii tunneling we discuss the leading asymptotics of eigenvalue splittings of 2 x2v in the limit as x o and where v is a non negative potential with several zeros for example if eo x e1 x are the two lowest eigenvalues in a situation where v has precisely two zeros a and b related by a symmetry then limx x lin e1 x eo x is given as the distance from a to b in a certain riemann metric

localization and coherence in nonintegrable systems we study the irreversible dynamics of nonlinear nonintegrable hamiltonian oscillator chains approaching their statistical asymptotic states in systems constrained by more than one conserved quantity the partitioning of the conserved quantities leads naturally to localized and coherent structures if the phase space is compact the final equilibrium state is governed by entropy maximization and the coherent structures are stable lumps in systems where the phase space is not compact the coherent structures can be collapsed represented in phase space by a heteroclinic connection of some unstable saddle to infinity

nekhoroshev theorem for small amplitude solutions in nonlinear schrodinger equations we prove a nekhoroshev type result 26 27 for the nonlinear schrodinger equation with vanishing or periodic boundary conditions on 0 pi here m in mathbb r is a parameter and varphi mathbb r to mathbb r is a function analytic in a neighborhood of the origin and such that varphi 0 0 varphi 0 not 0 more precisely we consider the cauchy problem for 0 1 with initial data which extend to analytic entire functions of finite order and prove that all the actions of the linearized system are approximate constants of motion up to times growing faster than any negative power of the size of the initial datum the proof is obtained by a method which applies to hamiltonian perturbations of linear pde s with the following properties i the linear dynamics is periodic ii there exists a finite order birkhoff normal form which is integrable and quasi convex as a function of the action variables eq 0 1 satisfies i and ii when restricted to a level surface of left vert u l 2 right vert which is an integral of motion the main technical tool used in the proof is a normal form lemma for systems with symmetry which is also proved here

a criterion of integrability for perturbed nonresonant harmonic oscillators wick ordering of the perturbations in classical mechanics and invariance of the frequency spectrum we introduce an analogue to the renormalization theory of quantum fields into classical mechanics we also find an integrability criterion guaranteeing the convergence of birkhoff s series and an algorithm for modifying the hamiltonian to fix the frequency spectrum of the quasi periodic motions we point out its possible relevance to the transition to chaos

time quasi periodic unbounded perturbations of schrodinger operators and kam methods we eliminate by kam methods the time dependence in a class of linear differential equations in l2 subject to an unbounded quasi periodic forcing this entails the pure point nature of the floquet spectrum of the operator h0 ep t for e small here h0 is the one dimensional schrodinger operator p2 v v x x 2 for x the time quasi periodic perturbation p may grow as x 2 2 and the frequency vector is non resonant the proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on kuksin s estimate of solutions of homological equations with non constant coefficients

nonautonomous discrete rogue wave solutions and interactions in an inhomogeneous lattice with varying coefficients abstract in this paper we analytically investigate nonautonomous discrete rogue wave solutions and their interactions in the generalized ablowitz ladik hirota lattice with variable coefficients which possess complicated wave propagation in time and differ from the usual discrete rogue waves when the amplitude of the tunnel coupling coefficient between sites decreases these nonautonomous discrete rogue wave solutions become localized in time after they propagate over some certain large critical values moreover we find that the interactions between nonautonomous discrete rogue wave solutions are elastic

spectral monodromy of non self adjoint operators in the present paper we build a combinatorial invariant called the spectral monodromy from the spectrum of a single non self adjoint h pseudodifferential operator with two degrees of freedom in the semi classical limit our inspiration comes from the quantum monodromy defined for the joint spectrum of an integrable system of n commuting self adjoint h pseudodifferential operators given by s vu ngoc quantum monodromy in integrable systems commun math phys 203 2 465 479 1999 the first simple case that we treat in this work is a normal operator in this case the discrete spectrum can be identified with the joint spectrum of an integrable quantum system the second more complex case we propose is a small perturbation of a self adjoint operator with a classical integrability property we show that the discrete spectrum in a small band around the real axis also has a combinatorial monodromy the main difficulty in this case is that we do not know the description of the spectrum everywher

bifurcation analysis of the localized modes dynamics in lattices with saturable nonlinearity abstract the dynamics of localized modes in discrete media with saturable nonlinearity are investigated the stability of stationary bright solitons is discussed from the view point of the energy minimum principle and mapping analysis due to the cascade saturation mechanism a bifurcation from trapped to transversely moving localized mode is found for particular values of high power the bifurcation coincides with the existence of the almost perfect separatrix in the corresponding area preserving map in addition the definition of the peierls nabarro effective potential is reconsidered

solitary wave dynamics in an external potential we study the behavior of solitary wave solutions of some generalized nonlinear schrodinger equations with an external potential the equations have the feature that in the absence of the external potential they have solutions describing inertial motions of stable solitary waves we consider solutions of the equations with a non vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that over a large interval of time they describe a solitary wave whose center of mass motion is a solution of newton s equations of motion for a point particle in the given external potential up to small corrections corresponding to radiation damping

nonlinear double well schrodinger equations in the semiclassical limit we consider time dependent schrodinger equations with a double well potential and an external nonlinear both local and non local perturbation in the semiclassical limit the finite dimensional eigenspace associated to the lowest eigenvalues of the linear operator is almost invariant for times of the order of the beating period and the dominant term of the wavefunction is given by means of the solutions of a finite dimensional dynamical system in the case of local nonlinear perturbation we assume the spatial dimension d 1 or d 2

a nonlinear schrodinger equation with two symmetric point interactions in one dimension we consider a time dependent one dimensional nonlinear schrodinger equation with a symmetric double well potential represented by two dirac s among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the hamiltonian then we establish the corresponding strichartz type estimate and we prove local existence and uniqueness of the solution to the original nonlinear problem

convergence of a quantum normal form and an exact quantization formula abstract the operator i on l 2 t l quantizing the linear flow of diophantine frequencies 1 l over t l l 1 is perturbed by the operator quantizing a function v x v x r l t l r z v z x r t l r real holomorphic the corresponding quantum normal form qnf is proved to converge uniformly in 0 1 this yields non trivial examples of quantum integrable systems an exact quantization formula for the spectrum and a convergence criterion for the birkhoff normal form valid for perturbations holomorphic away from the origin the main technical aspect concerns the solution of the quantum homological equation which is constructed and estimated by solving the moyal equation for the operator symbols the kam iteration can thus be implemented on the symbols and its convergence proved this entails the convergence of the qnf with radius estimated in terms only of the diophantine constants of

nash moser theory for standing water waves we consider a perfect fluid in periodic motion between parallel vertical walls above a horizontal bottom and beneath a free boundary at constant atmospheric pressure gravity acts vertically downwards suppose the underlying flow is two dimensional in a vertical plane orthogonal to the walls and satisfies the constant pressure condition on the free boundary where surface tension is neglected suppose also that it is symmetric about a plane midway between the walls and periodic in time such motion which can be extended to give a two dimensional flow of infinite horizontal extent that is periodic in space as well as in time is referred to as a standing wave unlike progressive or steady stokes waves standing waves are not stationary relative to a moving reference frame

growth of sobolev norms for time dependent periodic schrodinger equations with sublinear dispersion abstract in this paper we consider schrodinger equations with sublinear dispersion relation on the one dimensional torus t r 2 z more precisely we deal with equations of the form t u i v t u where v t is a quasi periodic in time self adjoint pseudo differential operator of the form v t v t x d m w t 0 m 1 d x x v is a smooth quasi periodic in time function and w is a quasi periodic time dependent pseudo differential operator of order strictly smaller than m under suitable assumptions on v and w we prove that if satisfies some non resonance conditions the solutions of the schrodinger equation t u i v t u grow at most as t t for any 0 the proof is based on a reduction to constant coefficients up to smoothing remainders of the vector field i v t which uses egorov type theorems and pseudo differential calculus the homological equations arising in the reduction procedure involve both time and space derivatives since the dispersion relation is sublinear such equations can be solved by imposing some melnikov non resonance conditions on the frequency vector

statistics of vector manakov rogue waves we present a statistical analysis based on the height and return time probabilities of high amplitude wave events in both focusing and defocusing manakov systems we find that analytical rational or semirational solutions associated with extreme rogue wave rw structures are the leading high amplitude events in this system we define the thresholds for classifying an extreme wave event as a rw our results indicate that there is a strong relationship between the type of rw and the mechanism which is responsible for its creation initially high amplitude events originate from modulation instability upon subsequent evolution the interaction among these events prevails as the mechanism for rw creation we suggest a strategy for confirming the basic properties of different extreme events this involves the definition of proper statistical measures at each stage of the rw dynamics our results point to the need for redefining criteria for identifying rw events

a minimal energy tracking method for non radially symmetric solutions of coupled nonlinear schrodinger equations we aim at developing methods to track minimal energy solutions of time independent m component coupled discrete nonlinear schrodinger dnls equations we first propose a method to find energy minimizers of the 1 component dnls equation and use it as the initial point of the m component dnls equations in a continuation scheme we then show that the change of local optimality occurs only at the bifurcation points the fact leads to a minimal energy tracking method that guides the choice of bifurcation branch corresponding to the minimal energy solution curve by combining all these techniques with a parameter switching scheme we successfully compute a non radially symmetric energy minimizer that can not be computed by existing numerical schemes straightforwardly

degenerate asymptotic perturbation theory asymptotic rayleigh schrodinger perturbation theory for discrete eigenvalues is developed systematically in the general degenerate case for this purpose we study the spectral properties ofm m matrix functionsa of a complex variable which have an asymptotic expansion ea k k as 0 we show that asymptotic expansions for groups of eigenvalues and for the corresponding spectral projections ofa can be obtained from the set a by analytic perturbation theory special attention is given to the case wherea is borel summable in some sector originating from 0 with opening angle here we prove that the asymptotic series describe individual eigenvalues and eigenprojections ofa which are shown to be holomorphic ins near 0 and borel summable ifa k a k for allk we then fit these results into the scheme of rayleigh schrodinger perturbation theory and we give some examples of asymptotic estimates for schrodinger operators

convergence versus integrability in birkhoff normal form we show that any analytically integrable hamiltonian system near an equilibrium point admits a convergent birkhoff normalization the proof is based on a new geometric approach to the topic

skew hermitian based iterations for nine point approximations of convection diffusion problems we study skew hermitian based splitting methods for the iterative solution of nonsymmetric linear systems arising from high order compact hoc approximations of two dimensional convection diffusion problems such discretisations lead to nine point system matrices with block tridiagonal structures the nonsymmetric discretisation matrix for a constant coefficient grid aligned flow problem is shown to be positive definite and thus hermitian and skew hermitian splitting hss methods can be considered for the solution of the corresponding linear system for the solution of this linear system using the hss iteration we derive an analytical expression for the optimal value for the upper bound of the contraction factor comparison of the hss iteration with triangular and skew hermitian splitting tss methods and their corresponding block variants btss are carried out for various test problems cpu timings for the different splitting methods using experimentally determined optimal acceleration parameters are also given our results show that the combination of high order compact discretisations and skew hermitian based iterative methods for solving the corresponding linear systems provide efficient procedures for approximating the solutions of two dimensional convection diffusion equations

spectral asymptotics via the semiclassical birkhoff normal form this article gives a simple treatment of the quantum birkhoff normal form for semiclassical pseudo differential operators with smooth coefficients the normal form is applied to describe the discrete spectrum in a generalised non degenerate potential well yielding uniform estimates in the energy e this permits a detailed study of the spectrum in various asymptotic regions of the parameters e h and gives improvements and new proofs for many of the results in the field in the completely resonant case we show that the pseudo differential operator can be reduced to a toeplitz operator on a reduced symplectic orbifold using this quantum reduction new spectral asymptotics concerning the fine structure of eigenvalue clusters are proved in the case of polynomial differential operators a combinatorial trace formula is obtained

mean field and classical limit of many body schrodinger dynamics for bosons we present a new proof of the convergence of the n particle schrodinger dynamics for bosons towards the dynamics generated by the hartree equation in the mean field limit for a restricted class of two body interactions we obtain convergence estimates uniform in hbar up to an exponentially small remainder for hbar 0 the classical dynamics in the mean field limit is given by the vlasov equation

krein s formula and one dimensional multiple well abstract it is shown that all the discrete eigenvalues of a one dimensional schrodinger operator with a multiple well potential possess an asymptotic expansions in power of h 1 2 when h 0 a formula for all the coefficients of these expansions is given the method uses two main tools perturbation by boundary conditions and exponential decay of eigenfunctions which are developed in this article as a by product of this work the exponential localization of eigenvectors when h goes to zero can be proved

exponentially accurate semiclassical asymptotics of low lying eigenvalues for 2 2 matrix schrodinger operators abstract we consider a simple molecular type quantum system in which the nuclei have one degree of freedom and the electrons have two levels the hamiltonian has the form h 4 2 2 y 2 h y where h y is a 2 2 real symmetric matrix near a local minimum of an electron level e y that is not at a level crossing we construct quasimodes that are exponentially accurate in the square of the born oppenheimer parameter by optimal truncation of the rayleigh schrodinger series that is we construct e and such that o 1 and h e exp 2 where 0

bounds on the tight binding approximation for the gross pitaevskii equation with a periodic potential abstract we justify the validity of the discrete nonlinear schrodinger equation for the tight binding approximation in the context of the gross pitaevskii equation with a periodic potential both piecewise constant and smooth potentials are considered in the semi classical limit while justification of stationary equations is developed in our previous work pelinovsky et al 2008 11 this work deals with time dependent space decaying solutions on large but finite time intervals

kam for quasi linear and fully nonlinear forced perturbations of airy equation we prove the existence of small amplitude quasi periodic solutions for quasi linear and fully nonlinear forced perturbations of the linear airy equation for hamiltonian or reversible nonlinearities we also prove their linear stability the key analysis concerns the reducibility of the linearized operator at an approximate solution which provides a sharp asymptotic expansion of its eigenvalues for quasi linear perturbations this cannot be directly obtained by a kam iteration hence we first perform a regularization procedure which conjugates the linearized operator to an operator with constant coefficients plus a bounded remainder these transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudo differential operators at this point we implement a nash moser iteration with second order melnikov non resonance conditions which completes the reduction to constant coefficients

on reducibility of schrodinger equations with quasiperiodic in time potentials we prove that a linear d dimensional schrodinger equation with an x periodic and t quasiperiodic potential reduces to an autonomous equation for most values of the frequency vector the reduction is made by means of a non autonomous linear transformation of the space of x periodic functions this transformation is a quasiperiodic function of t

exponentially accurate error estimates of quasiclassical eigenvalues ii several dimensions we study the behavior of truncated rayleigh schrodinger series for low lying eigenvalues of the time independent schrodinger equation in the semiclassical limit 0 in particular we prove that if the potential energy satisfies certain conditions there is an optimal truncation of the series for the eigenvalues in the sense that this truncation is exponentially close to the exact eigenvalue these results were already discussed for the one dimensional case in a previous article this time we consider the multi dimensional problem where degeneracy plays a central role

semiclassical dynamics with exponentially small error estimates we construct approximate solutions to the time dependent schrodinger equation for small values of if v satisfies appropriate analyticity and growth hypotheses and these solutions agree with exact solutions up to errors whose norms are bounded by for some c and 0 under more restrictive hypotheses we prove that for sufficiently small t implies the norms of the errors are bounded by for some c 0 and 0

the schrodinger equation and canonical perturbation theory lett0 ev be the schrodinger operator corresponding to the classical hamiltonianh0 ev whereh0 is thed dimensional harmonic oscillator with non resonant frequencies 1 d and the potentialv q1 qd is an entire function of order d 1 1 we prove that the algorithm of classical canonical perturbation theory can be applied to the schrodinger equation in the bargmann representation as a consequence each term of the rayleigh schrodinger series near any eigenvalue oft0 admits a convergent expansion in powers of of initial point the corresponding term of the classical birkhoff expansion moreover ifv is an even polynomial the above result and the kam theorem show that all eigenvalues n e oft0 ev such thatn coincides with a kam torus are given up to order e by a quantization formula which reduces to the bohr sommerfeld one up to first order terms in

a reduction theorem for time dependent schrodinger operator with finite differentiable unbounded perturbation in this paper we consider the time dependent schrodinger operator with a temporal quasi periodic perturbation which is some function of class cl for any l 100 3n 2 1 by using the kam technique we establish a reduction theorem for the above operator and prove that the invariant tori of the nearly integrable hamiltonian system with unbounded small perturbation are linearly stable both for d d 1 and the limiting case d d 1

exercises in exact quantization the formalism of exact 1d quantization is reviewed in detail and applied to the spectral study of three concrete schrodinger hamiltonians d2 dq2 v q on the half line q 0 with a dirichlet or neumann condition at q 0 emphasis is put on the analytical investigation of the spectral determinants and spectral zeta functions with respect to singular perturbation parameters we first discuss the homogeneous potential v q qn as n versus its solvable n limit an infinite square well useful distinctions are established between regular and singular behaviours of spectral quantities various identities among the square well spectral functions are unravelled as limits of finite n properties the second model is the quartic anharmonic oscillator the zero energy spectral determinants det d2 dq2 q4 vq2 are explicitly analysed in detail revealing many special values algebraic identities between taylor coefficients and functional equations of a quartic type coupled to asymptotic v properties of airy type the third study addresses the potentials v q qn vqn 2 1 of even degree their zero energy spectral determinants prove computable in closed form and the generalized eigenvalue problems with v as spectral variable admit exact quantization formulae which are perfect extensions of the harmonic oscillator case corresponding to n 2 these results partly reflect the presence of quasi exactly solvable potentials in the family above

floquet hamiltonians with pure point spectrum we consider floquet hamiltonians of the type k f i partial t h 0 beta v omega t whereh0 a selfadjoint operator acting in a hilbert space has simple discrete spectrume1 0 for a given 0 t v t is 2 periodic andr times strongly continuously differentiable as a bounded operator on and are real parameters and the periodic boundary condition is imposed in time we show roughly that providedr is large enough small enough and non resonant then the spectrum ofkf is pure point the method we use relies on a successive application of the adiabatic treatment due to howland and the kam type iteration settled by bellissard and extended by combescure both tools are revisited adjusted and at some points slightly simplified

exponential times in the one dimensional gross pitaevskii equation with multiple well potential we consider the gross pitaevskii equation in 1 space dimension with a n well trapping potential we prove in the semiclassical limit that the finite dimensional eigenspace associated to the lowest n eigenvalues of the linear operator is slightly deformed by the nonlinear term into an almost invariant manifold mathcal m precisely one has that solutions starting on mathcal m or close to it will remain close to mathcal m for times exponentially long with the inverse of the size of the nonlinearity as heuristically expected the effective equation on mathcal m is a perturbation of a discrete nonlinear schrodinger equation we deduce that when the size of the nonlinearity is large enough then tunneling among the wells essentially disappears that is for almost all solutions starting close to mathcal m their restriction to each of the wells has norm approximatively constant over the considered time scale in the particular case of a double well potential we give a more precise result showing persistence or destruction of the beating motions over exponentially long times the proof is based on canonical perturbation theory surprisingly enough due to the gauge invariance of the system no non resonance condition is required

local smoothing property and strichartz inequality for schrodinger equations with potentials superquadratic at infinity abstract we study smoothing properties for time dependent schrodinger equations i u t 1 2 u v x u u 0 l 2 r n with potentials which satisfy v x o x m at infinity m 2 we show that the solution u t x is 1 m times differentiable with respect to x at almost all t r and explain that this is the result of the fact that the sojourn time of classical particles with energy in arbitrary compact set is less than ct 1 m during 0 t when is very large we also show strichartz s inequality with derivative loss for such potentials and give its application to nonlinear schrodinger equations

horseshoes for coupled discrete nonlinear schrodinger equations in this paper we study the spatial disorder of coupled discrete nonlinear schrodinger cdnls equations with piecewise monotone nonlinearities by the construction of horseshoes we show that the cdnls equation possesses a hyperbolic invariant cantor set on which it is topological conjugate to the full shift on n symbols the cdnls equation exhibits spatial disorder resulting from the strong amplitudes and stiffness of the nonlinearities in the system the complexity of the disorder is determined by the oscillations of the nonlinearities we then apply our results to cdnls equations with kerr like nonlinearity we shall also show some patterns of the localized solutions of the cdnls equation

linear adiabatic theory exponential estimates a general adiabatic expansion is written down the basic result is that under appropriate smoothness conditions the adiabatic estimations can be pushed up to exponential order the results imply exponential estimates not only for hamiltonians analytic in a neighbourhood of the real axis but also for hamiltonians which in an appropriate sense beong to gevrey classes

destruction of the beating effect for a non linear schrodinger equation we consider a non linear perturbation of a symmetric double well potential as a model for molecular localization in the semiclassical limit we prove the existence of a critical value of the perturbation parameter giving the destruction of the beating effect this value is twice the one corresponding to the first bifurcation of the fundamental state here we make use of a particular projection operator introduced by g nenciu in order to extend to an infinite dimensional space some known results for a two level system

invariant cantor manifolds of quasi periodic solutions for the derivative nonlinear schrodinger equation abstract this paper is concerned with the derivative nonlinear schrodinger equation with periodic boundary conditions i u t u x x i f x u u x 0 x t r 2 z where f is an analytic function of the form f x u u u 2 u f 4 x u u 0 r and f 4 x u u denotes terms of order at least four in u u we show the above equation possesses cantor families of smooth quasi periodic solutions of small amplitude the proof is based on an infinite dimensional kam theorem for unbounded perturbation vector fields

dynamics of self trapping in the discrete nonlinear schro dinger equation abstract we study dynamical aspects of the discrete nonlinear schrodinger equation dnls for chains of different sizes with periodic and open boundary conditions we focus on the occurrence of a self trapping transition in the different geometries the initial condition used is that which places the particle or power on one lattice site or nonlinear waveguide and the quantity studied is the time averaged probability for the particle to remain in that site we show that the self trapping transition in long chains occurs for parameter values not very different from that for very small clusters

convergent expansions for tunneling a new method to compute effects of tunneling in one dimensional multiple well is developed a tunneling parameter built with physical quantities is introduced to measure the splitting between eigenvalues due to tunneling these splittings are given by convergent series in term of this tunneling parameter for a wide class of double well

normal forms and quantization formulae we consider the schrodinger operator where as is gevrey of order and has a unique non degenerate minimum a quantization formula up to an error of order is obtained for all eigenvalues of q lying in any interval with a 1 and 0 0 for eigenvalues in 0 1 the error is of order the proof is based upon uniform nekhoroshev estimates on the quantum normal form constructed quantizing the lie transformation

quasi periodic solutions for fully nonlinear forced reversible schrodinger equations abstract in this paper we consider a class of fully nonlinear forced and reversible schrodinger equations and prove existence and stability of quasi periodic solutions we use a nash moser algorithm together with a reducibility theorem on the linearized operator in a neighborhood of zero due to the presence of the highest order derivatives in the non linearity the classic kam reducibility argument fails and one needs to use a wider class of changes of variables such as diffeomorphisms of the torus and pseudo differential operators this procedure automatically produces a change of variables well defined on the phase space of the equation which diagonalizes the operator linearized at the solution this gives the linear stability

a local quantum version of the kolmogorov theorem consider in l2 l the operator family h e p0 eq0 p0 is the quantum harmonic oscillator with diophantine frequency vector q0 a bounded pseudodifferential operator with symbol holomorphic and decreasing to zero at infinity and e then there exists e 0 with the property that if e e there is a diophantine frequency e such that all eigenvalues en e of h e near 0 are given by the quantization formula open image in new window where is an l multi index

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

diophantine tori and spectral asymptotics for nonselfadjoint operators we study spectral asymptotics for small nonselfadjoint perturbations of selfadjoint h pseudodifferential operators in dimension 2 assuming that the classical flow of the unperturbed part possesses several invariant lagrangian tori enjoying a diophantine property we get complete asymptotic expansions for all eigenvalues in certain rectangles in the complex plane in two different cases in the first case we assume that the strength e of the perturbation is o hd for some d 0 and is bounded from below by a fixed positive power of h in the second case e is assumed to be sufficiently small but independent of h and we describe the eigenvalues completely in a fixed h independent domain in the complex spectral plane

a quantum cherry theorem for perturbations of the plane rotator we consider on l2 t2 the schrodinger operator family l r with domain and action defined as d l h2 t2 l u 12 2 1 12 2 22 u i 1 1 2 2 u vu here r 1 2 1 2 are vectors of complex non real frequencies and v a pseudodifferential operator of order zero l represents the weyl quantization of the hamiltonian family l x 12 1 12 2 22 1 1 2 2 v x defined on the phase space r2 t2 where v x c2 r2 t2 r we prove the uniform convergence with respect to 0 1 of the quantum normal form which reduces to the classical one for 0 this result simultaneously entails an exact quantization formula for the quantum spectrum as well as a convergence criterion for the classical birkhoff normal form generalizing a well known theorem of cherry

long time stability of some small amplitude solutions in nonlinear schrodinger equations we consider small perturbations of the zakharov shabat nonlinear schrodinger equation on 0 with vanishing or periodic boundary conditions we prove a nekhoroshev type result for solutions starting in the neighbourhood in the h 1 topology of the majority of small amplitude finite dimensional invariant tori of the linearized system more precisely we will prove that along the considered solutions all the actions of the linearized system are approximatively constant up to times growing exponentially with the inverse of a suitable small parameter

a uniform quantum version of the cherry theorem consider in l 2 mathbb r 2 the operator family h epsilon p 0 hbar omega epsilon f 0 p 0 is the quantum harmonic oscillator with diophantine frequency vector f 0 a bounded pseudodifferential operator with symbol decreasing to zero at infinity in phase space and epsilon in mathbb c then there exist epsilon ast 0 independent of hbar and an open set gamma subset mathbb c 2 setminus mathbb r 2 such that if epsilon epsilon ast and omega in gamma the quantum normal form near p 0 converges uniformly with respect to hbar this yields an exact quantization formula for the eigenvalues and for hbar 0 the classical cherry theorem on convergence of birkhoff s normal form for complex frequencies is recovered

order h 4 difference methods for a class of singular two space elliptic boundary value problems abstract in this article we derive difference methods of o h4 for solving the system of two space nonlinear elliptic partial differential equations with variable coefficients having mixed derivatives on a uniform square grid using nine grid points we obtain two sets of fouth order difference methods one in the absence of mixed derivatives second when the coefficients of uxy are not equal to zero and the coefficients of uxx and uyy are equal there do not exist fourth order schemes involving nine grid points for the general case the method having two variables has been tested on two dimensional viscous incompressible steady state navier stokes model equations in polar coordinates the proposed difference method for scalar equation is also applied to the poisson s equation in polar coordinates some numerical examples are provided to illustrate the fourth order convergence of the proposed methods

bifurcation and stability for nonlinear schrodinger equations with double well potential in the semiclassical limit we consider the stationary solutions for a class of schrodinger equations with a symmetric double well potential and a nonlinear perturbation here in the semiclassical limit we prove that the reduction to a finite mode approximation give the stationary solutions up to an exponentially small term and that symmetry breaking bifurcation occurs at a given value for the strength of the nonlinear term the kind of bifurcation picture only depends on the nonlinearity power we then discuss the stability instability properties of each branch of the stationary solutions finally we consider an explicit one dimensional toy model where the double well potential is given by means of a couple of attractive dirac s delta pointwise interactions

reducibility of quantum harmonic oscillator on rd with differential and quasi periodic in time potential abstract we improve the results in the following two aspects by grebert and paturel in 28 and prove that a linear schrodinger equation on r d with harmonic potential x 2 and small t quasiperiodic potential as i u t u x 2 u e v t x u 0 t x r r d reduces to an autonomous system for most values of the frequency vector r n firstly we assume the potential v is only in c t n h s r d with large enough secondly the condition on s is better than 28 as a consequence any solution of such a linear pde is almost periodic in time and remains bounded in some suitable sobolev norms

quantum monodromy in integrable systems let p1 h pn h be a set of commuting self adjoint h pseudo differential operators on an n dimensional manifold if the joint principal symbol p is proper it is known from the work of colin de verdiere 6 and charbonnel 3 that in a neighbourhood of any regular value of p the joint spectrum locally has the structure of an affine integral lattice this leads to the construction of a natural invariant of the spectrum called the quantum monodromy we present this construction here and show that this invariant is given by the classical monodromy of the underlying liouville integrable system as introduced by duistermaat 9 the most striking application of this result is that all two degree of freedom quantum integrable systems with a focus focus singularity have the same non trivial quantum monodromy for instance this proves a conjecture of cushman and duistermaat 7 concerning the quantum spherical pendulum

birkhoff normal form for partial differential equations with tame modulus we prove an abstract birkhoff normal form theorem for hamiltonian partial differential equations pdes the theorem applies to semilinear equations with nonlinearity satisfying a property that we call tame modulus such a property is related to the classical tame inequality by moser in the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times we also develop a general scheme to apply the abstract theory to pdes in one space dimensions and we use it to study some concrete equations nonlinear wave nlw equation nonlinear schrodinger nls equation with different boundary conditions an application to an nls equation on the d dimensional torus is also given in all cases we deduce bounds on the growth of high sobolev norms in particular we get lower bounds on the existence time of solutions

convergence of a quantum normal form and a generalization of cherry s theorem we consider on the schrodinger operator family with the domain and action defined as follows where 1 2 is a vector of complex frequencies and v is a pseudodifferential operator of order zero he represents the weyl quantization of the hamiltonian family defined on the phase space where we prove the uniform convergence with respect to 0 1 of the quantum normal form which reduces to the classical one for 0 as a consequence we simultaneously obtain an exact quantization formula for the quantum spectrum as well as a convergence criterion for the classical birkhoff normal form generalizing a well known theorem of cherry

standing waves on an infinitely deep perfect fluid under gravity the existence of two dimensional standing waves on the surface of an infinitely deep perfect fluid under gravity is established when formulated as a second order equation for a real valued function w on the 2 torus and a positive parameter the problem is fully nonlinear the highest order x derivative appears in the nonlinear term but not in the linearization at 0 and completely resonant there are infinitely many linearly independent eigenmodes of the linearization at 0 for all rational values of the parameter moreover for any prescribed order of accuracy there exists an explicit approximate solution of the nonlinear problem in the form of a trigonometric polynomial using a nash moser method to seek solutions of the nonlinear problem as perturbations of the approximate solutions the existence question can be reduced to one of estimating the inverses of linearized operators at non zero points after changing coordinates these operators become first order and non local in space and second order in time after further changes of variables the main parts become diagonal with constant coefficients and the remainder is regularizing or quasi one dimensional in the sense of 22 the operator can then be inverted for two reasons first the explicit formula for the approximate solution means that restricted to the infinite dimensional kernel of the linearization at zero the inverse exists and can be estimated second the small divisor problems that arise on the complement of this kernel can be overcome by considering only particular parameter values selected according to their diophantine properties a parameter dependent version of the nash moser implicit function theorem now yields the existence of a set of unimodal standing waves on flows of infinite depth corresponding to a set of values of the parameter 1 which is dense at 1 unimodal means that the term of smallest order in the amplitude is cos x cos t which is one of many eigenfunctions of the completely resonant linearized problem

extreme events in discrete nonlinear lattices we perform statistical analysis on discrete nonlinear waves generated through modulational instability in the context of the salerno model that interpolates between the integrable ablowitz ladik al equation and the nonintegrable discrete nonlinear schrodinger equation we focus on extreme events in the form of discrete rogue or freak waves that may arise as a result of rapid coalescence of discrete breathers or other nonlinear interaction processes we find power law dependence in the wave amplitude distribution accompanied by an enhanced probability for freak events close to the integrable limit of the equation a characteristic peak in the extreme event probability appears that is attributed to the onset of interaction of the discrete solitons of the al equation and the accompanied transition from the local to the global stochasticity monitored through the positive lyapunov exponent of a nonlinear map

spectrum for quantum duffing oscillator and small divisor equation with large variable coefficient by establishing a new estimate of the solutions for the small divisor equation with large variable coefficients and using the kam technique we prove a reduction theorem that entails the pure point nature of the floquet spectrum of the quantum duffing oscillator with a small perturbation temporal quasi periodic with nonresonant frequencies 2010 wiley periodicals inc

pure point spectrum of the floquet hamiltonian for the quantum harmonic oscillator under time quasi periodic perturbations we prove that the 1 d quantum harmonic oscillator is stable under spatially localized time quasi periodic perturbations on a set of diophantine frequencies of positive measure this proves a conjecture raised by enss veselic in their 1983 paper ev in the general quasi periodic setting the motivation of the present paper also comes from construction of quasi periodic solutions for the corresponding nonlinear equation

exponentially accurate error estimates of quasiclassical eigenvalues we study the behaviour of truncated rayleigh schrodinger series for the low lying eigenvalues of the one dimensional time independent schrodinger equation in the semiclassical limit 0 under certain hypotheses on the potential v x we prove that for any given small 0 there is an optimal truncation of the series for the approximate eigenvalues such that the difference between an approximate and exact eigenvalue is smaller than exp c for some positive constant c we also prove the analogous results concerning the eigenfunctions

convergence or generic divergence of the birkhoff normal form we prove that the birkhoff normal form of hamiltorlian flows at a nonresonant singular point with given quadratic part is always convergent or generically divergent the same result is proved for the normalization mapping and any formal first integral

a new fourth order discretization for singularly perturbed two dimensional non linear elliptic boundary value problems abstract in this article we derive a new difference method of o h4 so called arithmetic average discretization for the solution of two dimensional non linear singularly perturbed elliptic partial differential equation of the form e uxx uyy f x y u ux uy 0 0 is a small parameter we also derive new methods of o h4 for the estimates of u n which are quite often of interest in many physical problems in all cases we require only 9 grid points and a single computational cell the main advantage of the proposed methods is that the methods are directly applicable to singular problems we do not require any special technique or modification to solve singular problems numerical results are provided to demonstrate the usefulness of the methods discussed

absolute continuity of the floquet spectrum for a nonlinearly forced harmonic oscillator we prove that the floquet spectrum of the time periodic schrodinger equation corresponding to a mildly nonlinear resonant forcing is purely absolutely continuous for suitably small

the smart blage algorithm for singularly perturbed 2d elliptic partial differential equations we discuss a new smart block alternating group explicit smart blage algorithm for solving the symmetric block linear systems derived from nine point fourth order method based on arithmetic average discretizations to solve the singularly perturbed two dimensional elliptic equations the smart blage algorithm is predicted to save time as the idea involved is to eliminate evaluating two similar terms appearing in the algorithm the error analysis of the smart blage algorithm is discussed briefly numerical tests are performed to demonstrate the usefulness of the proposed smart blage method and compared with the corresponding block successive over relaxation bsor method by considering two test problems