relationships among three kinds of finite dimensional integrable hamiltonian systems abstract a new type of finite dimensional integrable hamiltonian systems containing many restricted soliton flows are considered their relations with the generalized classical gaudin models and the generalized classical gaudin models with boundary are established as an application the relationships among three different restricted mkdv flows are exhibited

backlund transformations as exact integrable time discretizations for the trigonometric gaudin model we construct a two parameter family of backlund transformations for the trigonometric classical gaudin magnet the approach follows closely the one introduced by sklyanin and kuznetsov 1998 j phys a math gen 31 2241 51 in a number of seminal papers and takes advantage of the intimate relation between the trigonometric and the rational case as in the paper by hone kuznetsov and one of the authors or 2001 j phys a math gen 34 2477 90 the backlund transformations are presented as explicit symplectic maps starting from their lax representation the expected connection with the xxz heisenberg chain is established and the rational xxx case is recovered in a suitable limit it is shown how to obtain a physical transformation mapping real variables into real variables the interpolating hamiltonian flow is derived and some numerical iterations of the map are presented

a new approach to inverse spectral theory i fundamental formalism we present a new approach distinct from gelfand levitan to the the orem of borg marchenko that the m function equivalently spectral measure for a finite interval or half line schrodinger operator determines the poten tial our approach is an analog of the continued fraction approach for the moment problem we prove there is a representation for the m function m 2 r b

quadratic operator relations and bethe equations for spin 1 2 richardson gaudin models in this work we demonstrate how one can in a generic approach derive a set of n simple quadratic bethe equations for integrable richardson gaudin rg models built out of n spins 1 2 these equations depend only on the n eigenvalues of the various conserved charges so that any solution of these equations defines indirectly through the corresponding set of eigenvalues one particular eigenstate the proposed construction covers the full class of integrable rg models of the xyz including the subclasses of xxz and xxx models type realised in terms of spins 1 2 coupled with one another through terms including as well magnetic field like terms linear in the pauli matrices the approach exclusively requires integrability defined here only by the requirement that n conserved charges r i with such that exist the result is therefore valid and equally simple for models with or without symmetry with or without a properly defined pseudo vacuum as well as for models with non skew symmetric couplings

formal solutions of semilinear heat equations abstract we study formal power series solutions to the initial value problem for semilinear heat equation t u u f u with polynomial nonlinearity f and prove that they belong to the formal gevrey class g 2 next we give counterexamples showing that the solution in general is not analytic in time at t 0

krein type theorems and ordered structure for cauchy de branges spaces abstract we extend some results of m g krein to the class of entire functions which can be represented as ratios of discrete cauchy transforms in the plane as an application we obtain new versions of de branges ordering theorem for nearly invariant subspaces in a class of hilbert spaces of entire functions examples illustrating sharpness of the obtained results are given we also discuss applications to spectral theory of rank one perturbations of normal operators

summability of divergent solutions of the n dimensional heat equation abstract the cauchy problem for n dimensional complex heat equation is considered the borel summability of formal solutions is characterized in terms of analytic continuation with an appropriate growth condition of the spherical mean of the cauchy data

on q asymptotics for linear q difference differential equations with fuchsian and irregular singularities abstract we consider a cauchy problem for some family of linear q difference differential equations with fuchsian and irregular singularities that admit a unique formal power series solution in two variables x t z for given formal power series initial conditions under suitable conditions and by the application of certain q borel and laplace transforms introduced by j p ramis and c zhang we are able to deal with the small divisors phenomenon caused by the fuchsian singularity and to construct actual holomorphic solutions of the cauchy problem whose q asymptotic expansion in t uniformly for z in the compact sets of c is x t z the small divisors effect is an increase in the order of q exponential growth and the appearance of a power of the factorial in the corresponding q gevrey bounds in the asymptotics

schrodinger operators and de branges spaces abstract we present an approach to de branges s theory of hilbert spaces of entire functions that emphasizes the connections to the spectral theory of differential operators the theory is used to discuss the spectral representation of one dimensional schrodinger operators and to solve the inverse spectral problem

classical functional bethe ansatz for sl n separation of variables for the magnetic chain it is desirable for one to be able to construct separation variables for systems with sl n r matrices or lax pairs this article relates and extends the two known methods of construction the sklyanin s functional bethe ansatz fba commun math phys 150 181 191 1992 was previously only known for sl 2 and sl 3 and associated r matrices the algebraic geometric method of adams harnad and hurtubise commun math phys 155 385 413 1993 has been shown to work for sl n but up until now only for linear poisson brackets in this article sklyanin s program is advanced by giving the fba for certain systems with sl n r matrices this is achieved by constructing rational functions a u and b u of the matrix elements of t u so that in the generic case the zeros xi of b u are the separation coordinates and the pi a xi provide their conjugate momenta it is shown that the separation variables thus defined are the same as those given as the simultaneous solutions of several equations in two

on integrable models related to the osp 1 2 gaudin algebra the osp 1 2 gaudin algebra is defined and integrable models described by it are considered the models include the osp 1 2 gaudin magnet and the dicke model related to it detailed discussion of the simplest cases of these models is presented the effect of the presence of fermions on the separation of variables is indicated

quadrics on real riemannian spaces of constant curvature separation of variables and connection with gaudin magnet the integrable systems are considered which are connected with separation of variables in real riemannian spaces of constant curvature an isomorphism is given for these systems with hyperbolic gaudin magnet using this isomorphism the complete classification of separable coordinate systems is provided by means of the corresponding l operators for the gaudin magnet

asymptotic existence theorems for formal power series whose coefficients satisfy certain partial differential recursions we study power series whose coefficients are holomorphic functions of another complex variable and a nonnegative real parameter s and are given by a differential recursion equation for positive integer s series of this form naturally occur as formal solutions of some partial differential equations with constant coefficients while for s 0 they satisfy certain perturbed linear ordinary differential equations for arbitrary s 0 these series solve a differential integral equation such power series in general are not multisummable however we shall prove existence of solutions of the same differential integral equation that in sectors of in general maximal opening have the formal series as their asymptotic expansion furthermore we shall indicate that the solutions so obtained can be related to one another in a fairly explicit manner thus exhibiting a stokes phenomenon

the asymptotic form of the titchmarsh weyl m function associated with a second order differential equation with locally integrable coefficient on deduit un developpement asymptotique pour la m fonction de titchmarsh weyl associee a l equation differentielle lineaire d ordre 2 y qy y sur 0 dans le cas ou la seule restriction sur la fonction a valeur reelle q est q l 1 0 e pour e 0

summability in general carleman ultraholomorphic classes a definition of summability is put forward in the framework of general carleman ultraholomorphic classes in sectors so generalizing k summability theory as developed by j p ramis departing from a strongly regular sequence of positive numbers we construct an associated analytic proximate order and corresponding kernels which allow us to consider suitable laplace and borel type transforms both formal and analytic whose behavior closely resembles that of the classical ones in the gevrey case an application to the study of the summability properties of the formal solutions to some moment partial differential equations is included

strongly regular sequences and proximate orders abstract summability methods for ultraholomorphic classes in sectors defined in terms of a strongly regular sequence m m p p n 0 have been put forward by a lastra s malek and the second author 28 we study several open questions related to the existence of kernels of summability constructed by means of analytic proximate orders in particular we give a simple condition that allows us to associate a proximate order with a strongly regular sequence under this assumption and through the characterization of strongly regular sequences in terms of so called regular variation we show that the growth index m defined by v thilliez 54 and the order of quasianalyticity m introduced by the second author 50 are the same

quadrics on complex riemannian spaces of constant curvature separation of variables and the gaudin magnet integrable systems that are connected with orthogonal separation of variables in complex riemannian spaces of constant curvature are considered herein an isomorphism with the hyperbolic gaudin magnet previously pointed out by one of the authors extends to coordinates of this type the complete classification of these separable coordinate systems is provided by means of the corresponding l matrices for the gaudin magnet the limiting procedures or e calculus which relate various degenerate orthogonal coordinate systems play a crucial role in the classification of all such systems

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

classical yang baxter equations and quantum integrable systems quantum integrable models associated with nondegenerate solutions of classical yang baxter equations related to the simple lie algebras are investigated these models are diagonalized for rational and trigonometric solutions in the cases of sl n gl n o n and sp n algebras the analogy with the quantum inverse scattering method is demonstrated

relaxation of nonlinear oscillations in bcs superconductivity the diagonal case of the sl 2 richardson gaudin quantum pairing model is known to be solvable as an abel jacobi inversion problem the effect of random time dependent perturbations of the single particle spectrum on the exact solution is an open question of considerable physical relevance weak perturbations introduce a new slow time scale while preserving the nonlinear character of the dynamics in this paper such perturbations are considered it is shown that the long time asymptotics can be obtained by a deformation of the original integrable system equivalent to phase averaging over the fast time scale

on the location of the weyl circles the paper deals with explicit estimates concerning certain circles in the complex plane which were associated with sturm liouville problems by h weyl by the use of riccati equations instead of linear integral equations improvements are obtained for results of everitt and halvorsen concerning the behaviour of the titchmarsh weyl m coefficient

integrable rosochatius deformations of the restricted soliton flows a method to construct integrable rosochatius deformations of the restricted soliton flows in the setup of lax formulation is presented the integrable rosochatius deformations of the restricted soliton flows such as the restricted ablowitz kaup newell segur flow the restricted tu meng flow the restricted tu flow with neumann type constraints and the restricted modified korteweg de vries flow together with their lax representations are presented in addition a lax representation of the jacobi rosochatius system is obtained

loop coproducts gaudin models and poisson coalgebras in this paper we show that if a is a poisson algebra equipped with a set of maps i a a n satisfying suitable conditions then the images of the casimir functions of a under the maps i that we call loop coproducts are in involution rational trigonometric and elliptic gaudin models can be recovered as particular cases of this construction and we show that the same happens for the integrable or partially integrable models that can be obtained through the so called coproduct method on the other hand we show that the loop coproduct approach provides a natural generalization of the gaudin algebras from the lie poisson to the generic poisson algebra context and hopefully can lead to the definition of new integrable models

on the relationship between classical gaudin models and bc type gaudin models two invertible transformations of parameters and variables which connect classical sl 2 gaudin models with bc type gaudin models are proposed the explicit corresponding formulae of the lax matrices and the r matrix relations are exhibited as applications we establish a relation of two kinds of restricted kaup newell flows present the bc type counterpart of the well known restricted akns flow and obtain the gaudin model corresponding to the hikami system

on k summability of formal solutions for a class of partial differential operators with time dependent coefficients we study the k summability of divergent formal solutions for the cauchy problem of a certain class of linear partial differential operators with time dependent coefficients the problem is reduced to a k summability property of formal solutions for a linear similar ordinary differential equation associated with the cauchy problem

borel summability of divergent solutions for singular first order partial differential equations with variable coefficients part i abstract under a certain restriction singular first order linear partial differential equations of nilpotent type with two variables are divided into two classes in the previous paper part i we dealt with the one class and comprehended that there was a close affinity between the borel summability of divergent solutions and global analytic continuation properties for coefficients in this part ii we give a similar consideration on the other class more precise global estimates than those given in part i for coefficients will be required to prove the borel summability of divergent solutions

separation of variables in the classical integrable sl 3 magnetic chain there are two fundamental problems studied by the theory of hamiltonian integrable systems integration of equations of motion and construction of action angle variables a third problem however should be added to the list separation of variables though much simpler thant the two others it has important relations to quantum integrability separation of variables is constructed for thesl 3 magnetic chain an example of an integrable model associated to a nonhyperelliptic algebraic curve

inverse spectral theory for one dimensional schrodinger operators the a function we link recently developed approaches to the inverse spectral problem due to simon and myself respectively we obtain a description of the set of simon s a functions in terms of a positivity condition this condition also characterizes the solubility of simon s fundamental equation

from integrable to superintegrable it is shown that the presence of multiple eigenvalue parameters may result in the superintegrability of the restricted soliton flows the restricted akns flow the neumann system and the restricted mkdv flow with an eigenvalue parameter whose multiplicity is greater than two are demonstrated to be superintegrable

a new approach to inverse spectral theory ii general real potentials and the connection to the spectral measure we continue the study of the a amplitude associated to a half line schrodinger operator d 2 dx 2 q in l 2 0 b b a is related to the weyl titchmarsh m function via m k 2 k a 0 a e 2 k d o e 2 k for all 0 we discuss five issues here first we extend the theory to general q in l 1 0 for all a including q s which are limit circle at infinity second we prove the following relation between the a amplitude and the spectral measure p a 2 1 2 sin 2 dp since the integral is divergent this formula has to be properly interpreted third we provide a laplace transform representation for m without error term in the case b fourth we discuss m functions associated to other boundary conditions than the dirichlet boundary conditions associated to the principal weyl titchmarsh m function finally we discuss some examples where one can compute a exactly

multisummability in carleman ultraholomorphic classes by means of nonzero proximate orders abstract we introduce a general multisummability theory of formal power series in carleman ultraholomorphic classes the finitely many levels of summation are determined by pairwise comparable nonequivalent weight sequences admitting nonzero proximate orders and whose growth indices are distinct thus we extend the powerful multisummability theory for finitely many gevrey levels developed by j p ramis j ecalle and w balser among others we provide both the analytical and cohomological approaches and obtain a reconstruction formula for the multisum of a multisummable series by means of iterated generalized laplace like operators

darboux coordinates and liouville arnold integration in loop algebras darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part tilde mathfrak g of loop algebras these are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves defined within a given matrix representation a liouville generating function is obtained in completely separated form and shown through the liouvile arnold integration method to lead to the abel map linearization of all hamiltonian flows induced by the spectral invariants as illustrative examples the caseg sl 2 together with its real forms is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics with the liouville generating function expressed in hyperellipsoidal coordinates forg sl 3 the method is applied to the computation of quasi periodic solutions of the two component coupled nonlinear schrodinger equation a case which requires further symplectic constraints in order to deal with singularities in the spectral data at

multisummability of formal solutions of some linear partial differential equations abstract let u t x n 1 un x tn be a formal power series in one variable t satisfying formally p t x t x u t x f t x t x c c d the purpose of the present paper is to introduce a class of linear partial differential operators and show that u t x is multisummable provided p t x t x belongs to this class

solvability of a class of integro differential equations and connections to one dimensional inverse problems abstract a new approach to one dimensional inverse problem was recently introduced by barry simon we continue the study on an intermediate object a which satisfies a nonlinear integro differential equation we prove local solvability of this a equation and find a necessary condition for global solvability some exact solutions are presented

gaudin magnet with boundary and generalized knizhnik zamolodchikov equation we consider the boundary quantum inverse scattering method established by sklyanin 1988 the gaudin magnet with boundary is diagonalized by taking a quasi classical limit of the inhomogeneous lattice using the method proposed by babujian 1993 the integral representation for the solution of the b type knizhnik zamolodchikov equation is explicitly constructed

borel summability of the heat equation with variable coefficients abstract we prove borel summability in nonsingular directions of solutions of partial differential equations of the form u t a z u z z where a z is a quartic polynomial and the initial condition is analytic in the special case a z z we obtain the detailed resurgent structure even in such a simple case the structure of the singular manifolds is quite intricate

on gevrey solutions of threefold singular nonlinear partial differential equations abstract we study gevrey asymptotics of the solutions to a family of threefold singular nonlinear partial differential equations in the complex domain we deal with both fuchsian and irregular singularities and allow the presence of a singular perturbation parameter by means of the borel laplace summation method we construct sectorial actual holomorphic solutions which turn out to share a same formal power series as their gevrey asymptotic expansion in the perturbation parameter this result rests on the malgrange sibuya theorem and it requires to prove that the difference between two neighboring solutions is exponentially small what in this case involves an asymptotic estimate for a particular dirichlet like series

multisummability of formal solutions to the cauchy problem for some linear partial differential equations abstract the paper considers the cauchy problem for linear partial differential equations of non kowalevskian type in the complex domain it is shown that if the cauchy data are entire functions of a suitable order the problem has a formal solution which is multisummable the precise bound of the admissible order of entire functions is described in terms of the newton polygon of the equation

separation of variables in the bc type gaudin magnet an integrable system is introduced based on the poisson rs matrix structure this is a generalization of the gaudin magnet and in the sl 2 case is isomorphic to the generalized neumann model the separation of variables is discussed for both the classical and quantum cases