properly embedded minimal planar domains in 1997 collin proved that any properly embedded minimal surface in r 3 with nite topology and more than one end has

birkhoff coordinates for the focusing nls equation in this paper we construct birkhoff coordinates for the focusing nonlinear schrodinger equation near the zero solution

a characterization of all elliptic algebro geometric solutions of the akns hierarchy here 9 x p x wl w3 denotes the elliptic weierstrass function with fundamental periods 2wl and 2w3 im w3 wl 0 in the special case where wl is real and w3 is purely imaginary the potential q x in 1 1 is real valued and ince s striking result 51 in modern spectral theoretic terminology yields that the spectrum of the unique self adjoint operator associated with the differential expression l2 d2 dx 2 q x in l2 r exhibits finitely many bands and gaps respectively that is

an example of blow up for the complex kdv equation and existence beyond the blow up the kdv equation with smooth complex initial data that blows up in a finite amount of time is studied first examples are given using the elliptic solutions of airault mckean and moser second the kdv flow of spectral coordinates is described showing that blow up occurs when eigenvalues run to infinity in finite time but eigenvalue collisions are harmless and do not cause a singularity eigenvalue collisions and blow up are studied numerically finally the solutions are continued through the singularity by continuing the coordinate flows on the appropriate riemann surface

analyticity of a multilayer mead markus plate abstract the multilayer mead markus plate consists of m 1 kirchhoff plate layers bound together by m shear deformable layers we consider the case in which the poisson ratios are the same in each kirchhoff plate layer and linear viscous shear damping is included in the shear deformable layers we show via a contradiction argument that the associated semigroup is analytic

elliptic solutions to difference non linear equations and related many body problems we study algebro geometric finite gap and elliptic solutions of fully discretized kp or 2d toda equations in bilinear form they are hirota s difference equation for functions starting from a given algebraic curve we express the function and the baker akhiezer function in terms of the riemann theta function we show that the elliptic solutions when the function is an elliptic polynomial form a subclass of the general algebro geometric solutions we construct the algebraic curves of the elliptic solutions the evolution of zeros of the elliptic solutions is governed by the discrete time generalization of the ruijsenaars schneider many body system the zeros obey equations which have the form of nested bethe ansatz equations known from integrable quantum field theories we discuss the lax representation and the action angle type variables for the many body system we also discuss elliptic solutions to discrete analogues of kdv sine gordon and 1d toda equations and describe the loci of the zeros

elliptic algebro geometric solutions of the kdv and akns hierarchies an analytic approach we provide an overview of elliptic algebro geometric solutions of the kdv and akns hierarchies with special emphasis on floquet theoretic and spectral theoretic methods our treatment includes an effective characterization of all stationary elliptic kdv and akns solutions based on a theory developed by hermite and picard

exact energy bands and fermi surfaces of separable abelian potentials we present a general theory for multidimensional schrodinger equations with separable abelian potentials with an arbitrary number of gaps in the spectrum in particular we derive general equations which allow one to express the energy and the wavevectors in the brillouin zone as a function of the spectral parameters by using the solutions of these equations we show how to construct the energy bands and the fermi surfaces in the first brillouin zone of the reciprocal lattice as illustrative examples we consider the case of two dimensional separable potentials with one two and three gaps in the spectrum the method can be applied to crystals with a cubic or a rectangular parallelogram wigner seitz cell in arbitrary dimensions the possibility to generalize the theory to other crystal symmetries is also briefly discussed

the nahm equations finite gap potentials and lame functions it is shown that the lame equation which is the simplest example of a finite gap hill s equation can be written in terms of the composition of two first order matrix operators the coefficients of which satisfy nahm s equation the characteristic eigenvalues of the lame equation emerge from the representation theory of the lie algebra so 3 a special case is that of reflectionless potentials and their associated bound states

on the shape of spectra for non self adjoint periodic schrodinger operators the spectra of the schrodinger operators with periodic potentials are studied when the potential is real and periodic the spectrum consists of at most countably many line segments energy bands on the real line while when the potential is complex and periodic the spectrum consists of at most countably many analytic arcs in the complex plane in some recent papers such operators with complex symmetric periodic potentials have been studied in particular the authors argued that some energy bands would appear and disappear under perturbations here we show that the appearance and disappearance of such energy bands imply existence of nonreal spectra this is a consequence of a more general result describing the local shape of the spectrum

elliptic solitons and grobner bases we consider the solution of spectral problems with elliptic coefficients in the framework of the hermite ansatz we show that the search for exactly solvable potentials and their spectral characteristics is reduced to a system of polynomial equations solvable by the grobner bases method and others new integrable potentials and corresponding solutions of the sawada kotera kaup kupershmidt boussinesq equations and others are found

picard potentials and hill s equation on a torus hill s equation has drawn an enormous amount of consideration due to its ubiquity in applications as well as its structural richness of particular importance in the last 20 years is its connection with the kdv hierarchy and hence with integrable systems we show in this paper that regarding the independent variable as a complex variable yields a breakthrough for the problem of an efficient characterization of all elliptic finitegap potentials a major open problem in the field specifically we show that elliptic finitegap potentials of hill s equation are precisely those for which all solutions for all spectral parameters are meromorphic functions in the independent variable complementing a classical theorem of picard the intimate connection between picard s theorem and elliptic finite gap solutions of completely integrable systems is established in this paper for the first time in addition we construct the hyperelliptic riemann surface associated with a finitegap potential not necessarily elliptic i e determine its branch and singular points from a comparison of the geometric and algebraic multiplicities of eigenvalues of certain auxiliary operators associated with hill s equation these multiplicities are intimately correlated with the pole structure of the diagonal green s function of the operator h d2 dx2 q x in l2 r our construction is new in the present general complex valued periodic finite gap case before describing our approach in some detail we shall give a brief account of the history of the problem involved this theme dates back to a 1940 paper of ince 43 who studied what is presently called the lam6 ince potential

the hermite krichever ansatz for fuchsian equations with applications to the sixth painleve equation and to finite gap potentials several results including integral representation of solutions and hermite krichever ansatz on heun s equation are generalized to a certain class of fuchsian differential equations and they are applied to equations which are related with physics we investigate linear differential equations that produce painleve equation by monodromy preserving deformation and obtain solutions of the sixth painleve equation which include hitchin s solution the relationship with finite gap potential is also discussed we find new finite gap potentials namely we show that the potential which is written as the sum of the treibich verdier potential and additional apparent singularities of exponents 1 and 2 is finite gap which extends the result obtained previously by treibich we also investigate the eigenfunctions and their monodromy of the schrodinger operator on our potential

trace formulas for non self adjoint periodic schrodinger operators and some applications abstract recently a trace formula for non self adjoint periodic schrodinger operators in l 2 r associated with dirichlet eigenvalues was proved in differential integral equations 14 2001 671 700 here we prove a corresponding trace formula associated with neumann eigenvalues in addition we investigate dirichlet and neumann eigenvalues of such operators in particular using the dirichlet and neumann trace formulas we provide detailed information on location of the dirichlet and neumann eigenvalues for the model operator with the potential k e 2 i x where k c

null controllability of von karman thermoelastic plates under the clamped or free mechanical boundary conditions abstract in this paper we provide results of local and global null controllability for 2 d thermoelastic systems in the absence of rotational inertia and under the influence of the nonlipschitz von karman nonlinearity the plate component may be taken to satisfy either the clamped or higher order and physically relevant free boundary conditions in the accompanying analysis critical use is made of sharp observability estimates which obtain for the linearization of the thermoelastic plate these being derived in g avalos i lasiecka the null controllability of thermoelastic plates and singularity of the associated minimal energy function j math anal appl 294 2004 34 61 and g avalos i lasiecka asymptotic rates of blowup for the minimal energy function for the null controllability of thermoelastic plates the free case in proc of the conference for the control of partial differential equations georgetown university dekker in press moreover another key ingredient in our work to steer the given nonlinear dynamics is the recent result in a favini m a horn i lasiecka d tataru addendum to the paper global existence uniqueness and regularity of solution to a von karman system with nonlinear boundary dissipation differential integral equations 10 1997 197 200 concerning the sharp regularity of the von karman nonlinearity

rational solutions of the kadomtsev petviashvili hierarchy and the dynamics of their poles i new form of a general rational solution a new approach to the construction of the rational solutions to the hierarchy of the kadomtsev petviashvili equation is presented the generalization of the superposition formula is found to permit the construction of a new general solution from some partial ones the features of the general polynomial factorization and the types of scattering in the many body calogero moser problem are investigated

sharp polynomial bounds on the number of scattering poles of radial potentials abstract it is shown that for scattering by a radially symmetric potential in r n n odd the number of poles in a disk of radius r satisfies an estimate n r c n r 1 n this bound is sharp as shown by the special case of potentials nonvanishing at the boundary where n r k n a n r n 1 o 1 a being the diameter of the support

on the two gap locus for the elliptic calogero moser model we give an analytical description of the locus of the two gap elliptic potentials associated with the corresponding flow of the calogero moser system we start with the description of treibich verdier two gap elliptic potentials the explicit formulae for the covers wavefunctions and lame polynomials are derived together with a new lax representation for the particle dynamics on the locus we then consider more general potentials within the weierstrass reduction theory of theta functions to lower genera the reduction conditions in the moduli space of the genus 2 algebraic curves are given this is a subvariety of the humbert surface which can be singled out by the condition of the vanishing of some theta constants

sliding order and sliding accuracy in sliding mode control the synthesis of a control algorithm that stirs a nonlinear system to a given manifold and keeps it within this constraint is considered usually what is called sliding mode is employed in such synthesis this sliding mode is characterized in practice by a high frequency switching of the control it turns out that the deviation of the system from its prescribed constraints sliding accuracy is proportional to the switching time delay a new class of sliding modes and algorithms is presented and the concept of sliding mode order is introduced these algorithms feature a bounded control continuously depending on time with discontinuities only in the control derivative it is also shown that the sliding accuracy is proportional to the square of the switching time delay

the symmetries of solitons in this article we will retrace one of the great mathematical adventures of this century the discovery of the soliton and the gradual explanation of its remarkable properties in terms of hidden symmetries we will take an historical approach starting with a famous numerical experiment carried out by fermi pasta and ulam on one of the first electronic computers and with zabusky and kruskal s insightful explanation of the surprising results of that experiment and of a follow up experiment of their own in terms of a new concept they called solitons solitons however raised even more questions than they answered in particular the evolution equations that govern solitons were found to be hamiltonian and have infinitely many conserved quantities pointing to the existence of many non obvious symmetries we will cover next the elegant approach to solitons in terms of the inverse scattering transform and lax pairs and finally explain how those ideas led step by step to the discovery that loop groups acting by dressing transformations give a conceptually satisfying explanation of the secret soliton symmetries

isophasal isopolar and isospectral schrodinger operators and elementary complex analysis we explicitly construct super exponentially decaying potentials on d so that the associated schrodinger operators are isophasal and isopolar we use similar techniques to construct isospectral schrodinger operators with periodic potentials on and isospectral schrodinger operators on

asymptotics for a resonance counting function for potential scattering on cylinders abstract we study certain resonance counting functions for potential scattering on infinite cylinders or half cylinders under certain conditions on the potential we obtain asymptotics of the counting functions with an explicit formula for the constant appearing in the leading term

spectral problems of a class of non self adjoint one dimensional schrodinger operators abstract in this paper we investigate the one dimensional schrodinger operator l q with complex valued periodic potential q when q l 1 0 1 and q n 0 for n 0 1 2 where q n are the fourier coefficients of q with respect to the system e i 2 n x we prove that the bloch eigenvalues are 2 n t 2 for n z t c and find explicit formulas for the bloch functions then we consider the inverse problem for this operator

quasi periodic solutions of mixed akns equations abstract a hierarchy of new nonlinear evolution equations which are composed of the positive and negative akns flows is proposed on the basis of the theory of algebraic curves the corresponding flows are straightened using the abel jacobi coordinates the meromorphic function the baker akhiezer vector and the hyperelliptic curve k n are introduced and by using these quasi periodic solutions of the first three nonlinear evolution equations in the hierarchy are constructed according to the asymptotic properties and the algebro geometric characters of and k n

the xi function despite the fact that spectral and inverse spectral properties of one dimensional schrsdinger operators h d2 dx2 have been extensively studied for seventy five years there remain large areas where our knowledge is limited for example while the inverse theory for operators on l2 co co is well understood in case v is periodic 12 24 25 35 39 42 49 it is not understood in case limjx o v x co and h has discrete spectrum our goal here is to introduce a special function x a on r associated to h which we believe will be a valuable tool in the spectral and inverse spectral theory in a sense we will make precise it complements the weyl m functions m x a a main application of which we will make here concerns a generalization of the trace formula for schrsdinger operators to general v s recall the well known trace formula for periodic potentials let v x v x t 1 then by floquet theory see e g 10 37 44

on hill s equation with a singular complex valued potential 1 introduction the differential equation y00 qy ey is called hill s equation if q is a realvalued periodic locally integrable function of a real variable it is very well known that the spectrum of the operator associated with this equation consists of a countably in nite number of compact intervals of the real line these intervals are called spectral bands and are generally separated by open intervals called gaps in some instances however all but a nite number of gaps are empty resulting in a spectrum consisting of a nite number of compact bands and one closed interval extending to negative in nity in this case one calls q a nite band or nite gap potential the relationship of nite band potentials with the hierarchy of korteweg de vries equations has been the reason for a heightened interest in them during the past twenty years for recent accounts see for instance the monographs by dickey 8 and belokolos et al 3 the rst non trivial example of a nite band potential was given by ince in 1940 18 consider the lame potentials q x yg g 1 x c jq q0 where g 2n and jq q0 denotes weierstrass s elliptic function with fundamental half periods q and q0 if q is real q0 is pure imaginary and c q0 then q is realvalued periodic and continuous ince s result when translated to the present language shows that q then is a nite band potential with g non empty gaps between the bands this however raises immediately the question of what happens when c 6 q0 or when the half periods are more general in these cases one encounters potentials which are complex valued but continuous as well as potentials which have singularities of the form yg g 1 x and are realor complex valued before giving an overview of the content of the paper i want to give a brief account of the history of the subject let q be a real locally integrable periodic function with period p l the differential expression d dx q and h the self adjoint operator in l r associated with l then consider the following four statements a there are precisely 2g 1 real numbers e2g e0 such that the spectrum of h is given by

quasi periodic and periodic solutions for coupled nonlinear schrodinger equations of manakov type we consider travelling periodic and quasi periodic wave solutions in coupled nonlinear schrodinger equations in fibre optics these equations can be used to model single mode fibres with strong birefringence and two mode optical fibres recently these equations appear as a model describing pulse pulse interactions in wavelength division multiplexed channels of optical fibre transmission systems in some cases this model reduces to the integrable manakov system ims two phase quasi periodic solutions for the ims are given in terms of two dimensional kleinian functions the reduction of quasi periodic solutions to elliptic functions is discussed new solutions are found in terms of generalized hermite polynomials which are associated with two gap treibich verdier potentials

quasiperiodic and periodic solutions for vector nonlinear schrodinger equations we consider quasiperiodic and periodic cnoidal wave solutions of a set of n component vector nonlinear schrodinger equations vnlses in a biased photorefractive crystal with a drift mechanism of nonlinear response and kerr type nonlinearity n component nonlinear schrodinger equations can be used to model self trapped mutually incoherent wave packets these equations also model pulse pulse interactions in wavelength division multiplexed channels of optical fiber transmission systems quasiperiodic wave solutions for the vnlses in terms of n dimensional kleinian functions are presented periodic solutions in terms of hermite polynomials and generalized hermite polynomials for n component nonlinear schrodinger equations are found

complex hill s equation and the complex periodic korteweg de vries equations on etudie la classe spectrale des operateurs differentiels l d 2 dx 2 q avec un potentiel q de periode un on traite des potentiels complexes et des operateurs auto adjoints

lame equation sl 2 algebra and isospectral deformations it is proven that the n zone lame equation is equivalent to a spectral problem for a quadratic element of a universal enveloping sl 2 algebra the riemannian surface corresponding to the eigenstates in a parametric space forms the 2n 1 sheet surface which splits in four subsurfaces three of them contain the same number of sheets

on the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations the paper is concerned with the completeness problem of root functions of general boundary value problems for first order systems of ordinary differential equations we introduce and investigate the class of weakly regular boundary conditions we show that this class is much broader than the class of regular boundary conditions introduced by g d birkhoff and r e langer our main result states that the system of root functions of a boundary value problem is complete and minimal provided that the boundary conditions are weakly regular moreover we show that in some cases the weak regularity of boundary conditions is also necessary for the completeness also we investigate the completeness for 2 2 dirac type equations subject to irregular boundary conditions emphasize that our results are the first results on the completeness for general first order systems even in the case of regular boundary conditions

elliptic solutions and blow up in an integrable henon heiles system we consider an integrable case of the henon heiles system and use an isomorphism with the two gap kdv flow to construct families of real elliptic trajectories which are associated with two gap elliptic solitons of the kdv equation some of these solutions exhibit blow up in finite time

eigenvalues of regular sturm liouville problems abstract the eigenvalues of sturm liouville sl problems depend not only continuously but smoothly on the problem an expression for the derivative of an eigenvalue with respect to a given parameter an endpoint a boundary condition a coefficient or the weight function is found

linearization by output injection and nonlinear observers abstract observers can easily be constructed for those nonlinear systems which can be transformed into a linear system by change of state variables and output injection necessary and sufficient conditions for the existence of such a transformation are given

the classical theory of minimal surfaces we present here a survey of recent spectacular successes in classical minimal surface theory we highlight this article with the theorem that the plane the helicoid the catenoid and the one parameter family frtgt2 0 1 of riemann minimal examples are the only complete properly embedded minimal planar domains in r 3 the proof of this result depends primarily on work of colding

a schauder and riesz basis criterion for non self adjoint schrodinger operators with periodic and antiperiodic boundary conditions under the assumption that v l2 0 dx we derive necessary and sufficient conditions in terms of spectral data for non self adjoint schrodinger operators d2 dx2 v in l2 0 dx with periodic and antiperiodic boundary conditions to possess a riesz basis of root vectors i e eigenvectors and generalized eigenvectors spanning the range of the riesz projection associated with the corresponding periodic and antiperiodic eigenvalues we also discuss the case of a schauder basis for periodic and antiperiodic schrodinger operators d2 dx2 v in lp 0 dx p 1

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

schrodinger operators with complex valued potentials and no resonances in dimension d 3 we give examples of nontrivial compactly supported complex valued potentials such that the associated schrodinger operators have neither resonances nor eigenvalues if d 2 we show that there are potentials with no resonances or eigenvalues away from the origin these schrodinger operators are isophasal and have the same scattering phase as the laplacian on rd in odd dimensions d 3 we study the fundamental solution of the wave equation perturbed by such a potential if the space variables are held fixed it is superexponentially decaying in time

explicit quasi periodic solutions of the vakhnenko equation the trigonal curve associated with the vakhnenko equation is introduced by using the lax matrix for the nth stationary positive flow based on the theory of the trigonal curve and the properties of the three kinds of abel differentials the riemann theta function representation for the baker akhiezer function is derived from which the straightening out of flows in the jacobian variety is exactly given through the abel maps we finally arrive at quasi periodic solutions of the vakhnenko equation

criteria for existence of riesz bases consisting of root functions of hill and 1d dirac operators abstract we study the system of root functions srf of hill operator l y y v y with a singular complex valued potential v h per 1 and the srf of 1d dirac operator l y i 1 0 0 1 d y d x v y with matrix l 2 potential v 0 p q 0 subject to periodic or anti periodic boundary conditions series of necessary and sufficient conditions in terms of fourier coefficients of the potentials and related spectral gaps and deviations for srf to contain a riesz basis are proven equiconvergence theorems are used to explain basis property of srf in l p spaces and other rearrangement invariant function spaces

the null controllability of thermoelastic plates and singularity of the associated minimal energy function abstract we consider the null controllability problem for thermoelastic plates defined on a two dimensional domain and subject to hinged clamped or free boundary conditions the uncontrolled partial differential equation system generates an analytic semigroup on the space of finite energy consequently the concept of null controllability is indeed appropriate for consideration here it is shown that all finite energy states can be driven to zero by means of just one l 2 0 t control be it either mechanical or thermal the singularity as t 0 of the associated minimal energy function is the main object studied in the paper singularity and blow up rates for minimal energy function are not only of interest in their own right but are also of critical importance in stochastic pdes in this paper we establish the optimal blow up rate o t 5 2 for this function it is shown that the rate of singularity is the same as for finite dimensional truncations of the model in view of sharp estimates available in the finite dimensional setting math control signals systems 9 1997 327 the singularity rates provided in this paper are optimal

commuting linear differential expressions we consider the question when do two ordinary linear differential expressions commute it turns out that the set of all expressions which commute with a given one form a commutative ring here we study the algebraic structure of these rings as an application a complete characterization of normal differential expressions is obtained

exact solutions for a class of integrable henon heiles type systems we study the exact solutions of a class of integrable henon heiles type systems according to the analysis of bountis et al phys rev a 25 1257 1982 these solutions are expressed in terms of two dimensional kleinian functions special periodic solutions are expressed in terms of the well known weierstrass function we extend some of our results to a generalized henon heiles type system with n 1 degrees of freedom

smoothness of schrodinger operator potential in the case of gevrey type asymptotics of the gaps abstract consider the schrodinger equation y v x y y with a periodic real valued l 2 potential v of period 1 v x m v m exp 2 imx let n n be its zones of instability i e n n are pairs of periodic and antiperiodic eigenvalues and b 0 1 determines gevrey classes theorem if n 1 n n 2 1 n 2s e 2an b s 0 a0 then v x is a gevrey function and moreover m v m 2 1 m 2s e 2a m b

observability forms for switched systems with zeno phenomenon or high switching frequency this technical note deals with the observability of a class of switched systems with zeno phenomenon or high switching frequency particularly three observability forms are proposed and the observability for each form with knowledge of filtered switching signal is analyzed meanwhile sufficient and necessary conditions for the existence of a diffeomorphism to transform a class of switched systems into one of such forms are presented examples and simulations are given at the end to highlight the theoretical results

on a theorem of hochstadt in fact hochstadt gives an explicit formula for the discriminant which will be reproduced below since hochstadt s work there has been a considerable interest in finite band potentials i e potentials q such that the spectrum of the operator associated with d2 dx 2 q consists of finitely many regular analytic arcs due to their close relationship with the korteweg de vries hierarchy and hence with integrable systems however the interest is not restricted to real valued potentials for instance

determinants of laplacians and isopolar metrics on surfaces of infinite area we construct a determinant of the laplacian for infinite area surfaces which are hyperbolic near infinity and without cusps in the case of a convex co compact hyperbolic metric the determinant can be related to the selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the laplacian in the hyperbolic near infinity case the determinant is analyzed through the zeta regularized relative determinant for a conformal metric perturbation we establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics these results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances

on the instability tongues of the hill equation coupled with a conservative nonlinear oscillator abstract we study the asymptotics for the lengths l n q of the instability tongues of hill equations that arise as iso energetic linearization of two coupled oscillators around a single mode periodic orbit we show that for small energies i e q 0 the instability tongues have the same behavior that occurs in the case of the mathieu equation l n q o q n the result follows from a theorem which fully characterizes the class of hill equations with the same asymptotic behavior in addition in some significant cases we characterize the shape of the instability tongues for small energies motivation of the paper stems from recent mathematical works on the theory of suspension bridges

the spectral expansion for a nonself adjoint hill operator with a locally integrable potential abstract we construct the spectral expansion for the one dimensional schrodinger operator l d 2 dx 2 q x x in l 2 where q x is a 1 periodic lebesgue integrable on 0 1 and complex valued potential we obtain the asymptotic formulas for the eigenfunctions and eigenvalues of the operator l t for t 0 generated by this operation in l 2 0 1 and the t periodic boundary conditions using it we prove that the eigenfunctions and associated functions of l t form a riesz basis in l 2 0 1 for t 0 then we find the spectral expansion for the operator l

an explicit characterization of calogero moser systems combining theorems of halphen floquet and picard and a frobenius type analysis we characterize rational meromorphic simply periodic and elliptic kdv potentials in particular we explicitly describe the proper extension of the airault mckean moser locus associated with these three classes of algebro geometric solutions of the kdv hierarchy with special emphasis on the case of multiple collisions between the poles of solutions this solves a problem left open since the mid 1970s

on the integrable geometry of soliton equations and n 2 supersymmetric gauge theories we provide a unified construction of the symplectic forms which arise in the solution of both n 2 supersymmetric yang mills theories and soliton equations their phase spaces are jacobian type bundles over the leaves of a foliation in a universal configuration space on one hand imbedded into finite gap solutions of soliton equations these symplectic forms assume explicit expressions in terms of the auxiliary lax pair expressions which generalize the well known gardner faddeev zakharov bracket for kdv to a vast class of 2d integrable models on the other hand they determine completely the effective lagrangian and bps spectrum when the leaves are identified with the moduli space of vacua of an n 2 supersymmetric gauge theory for su n c with n f leq n c 1 flavors the spectral curves we obtain this way agree with the ones derived by hanany and oz and others from physical considerations

convergence of spectral decompositions of hill operators with trigonometric polynomial potentials we consider the hill operator ly y prime prime v x y quad0 leq x leq pi subject to periodic or antiperiodic boundary conditions with potentials v which are trigonometric polynomials with nonzero coefficients of the form then the system of eigenfunctions and at most finitely many associated functions is complete but it is not a basis in l 2 0 pi mathbb c if a b in the case i if a b and neither b 2 4b nor a 2 4a is an integer square in the case iii and it is never a basis in the case ii subject to periodic boundary conditions

elliptic baker akhiezer functions and an application to an integrable dynamical system an ansatz for elliptic baker akhiezer functions is used to find new solutions for various spectral problems the sturm liouville problem with four and five gap lame potentials and two third order problems one of the third order problems is the classical halphen equation with a spectral parameter the eigenfunctions are used to construct elliptic solutions for integrable cases of a generalized henon heiles system

spectral properties of a certain class of complex potentials in this paper we discuss spectral properties of the schroedinger operator a q on compact homogeneous spaces for certain complex valued potentials q we show for instance that for these potentials the spectrum of a q is identical with the spectrum of a introduction let g be a compact lie group and a the bi invariant laplacebeltrami operator on g fegan observed in f that there exist functions q on g for which the schroedinger operator q possesses an infinite sequence of eigenvalues al 2 such that each a is also an eigenvalue of a in this paper we will show that a potentials with this property exist not just on compact lie groups but on arbitrary compact homogeneous spaces as well and b for such potentials the entire spectrum of a q is identical with the spectrum of a in fegan s examples as well as in ours the potentials q are complex valued this means that the operator a q is not selfadjoint fortunately however there exists a rather nice spectral theory for such operators for instance their generalized eigenspaces are finite dimensional and span l2 see 2 below in view of this we were surprised to find relatively little material in the literature on such operators a notable exception is peter sarnack s beautiful paper on schroedinger operators with quasiperiodic q s see s we will describe and sketch the proof of our main result let g be a compact connected lie group h a closed subgroup and x the homogeneous space g h for simplicity we will assume for the moment that g is semisimple though this assumption is not essential see 4 for the case g s let g be the unitary dual of g we can order g lexicographically by choosing a system of positive weights consider the hilbert space direct sum 1 1 l2 x ha a eg ha being the g invariant subspace of l2 x on which g acts according to the representation a each irreducible subspace of ha possesses a unique maximal weight vector let h be the subspace of ha spanned by these vectors and let 1 2 l2 x ha received by the editors august 18 1982 and in revised form november 19 1982 1980 mathematics subject classification primary 58g25 secondary 35p99 research supported by the national university of mexico c1983 american mathematical society 0002 9947 83 1 00 25 per page

huygens principle for hyperbolic operators and integrable hierarchies abstract we show that the stationary solutions of the canonical akns hierarchy of nonlinear evolution equations yield perturbations of dirac operators that satisfy a strict form of huygens principle namely the domain of dependence of such dirac operators at any given point y is contained in the light cone s hypersurface issued from y by the canonical akns hierarchy we mean that the differential polynomials defining the flows are isobaric with respect to certain weights the method we employ is of interest by itself indeed we consider the riesz kernels associated to a given hyperbolic differential operator and expand the fundamental solution of perturbations of this operator in a series in such riesz kernels using the coefficients of this hadamard type expansion we introduce a family of vector fields for the d alembertian such vector field family corresponds to the kdv hierarchy and for the dirac operators they include the akns one

on tau functions for orthogonal polynomials and matrix models let v be a real polynomial of even degree and let rho be the equilibrium probability measure for v with support s to that v x greeater than or equal to int 2log x y rho dy c for some constant c with equality on s then s is the union of finitely many boundd intervals with endpoints delta j and rho is given by an algebraic weight w x on s then the system of orthogonal polynomials for w gives rise to a system of differential equations known as the schlesinger equations this paper identifies the tau function of this system with the hankel determinant det int x j k rho dx the solutions of the magnus schlesinger equation are realised by a linear system which is used to compute the tau functions in terms of a gelfand levitan equation the tau function is associated with a potential q and a scattering problem for the schrodinger equation with potential q the paper describes cases where this is integrable in terms of the nonlinear fourier transform

concerning the exact null controllability of a nonlinear thermoelastic plate abstract in this paper we consider the null controllability problem for a partial differential equation pde which models thin von karman plate dynamics in the absence of rotational inertia using the fact that thelinearization of this thermoelastic pde is null controllable we proceed to show that the full von karman dynamics are locally null controllable in obtaining the relevant observability inequalities key use is made of the abstract multiplier which is apparently intrinsic to control theoretic studies of thermoelastic plates

on a theorem of halphen and its application to integrable systems abstract we extend halphen s theorem which characterizes solutions of certain nth order differential equations with rational coefficients and meromorphic fundamental systems to a first order n n system of differential equations as an application of this circle of ideas we consider stationary rational algebro geometric solutions of the kdv hierarchy and illustrate some of the connections with completely integrable models of the calogero moser type in particular our treatment recovers the complete characterization of the isospectral class of such rational kdv solutions in terms of a precise description of the airault mckean moser locus of their poles

divergence of spectral decompositions of hill operators with two exponential term potentials abstract we consider the hill operator l y y v x y 0 x subject to periodic or antiperiodic boundary conditions bc with potentials of the form v x a e 2 i r x b e 2 i s x a b 0 r s n r s it is shown that the system of root functions does not contain a basis in l 2 0 c if bc are periodic or if bc are antiperiodic and r s are odd or r 1 and s 3

separation of variables for the ruijsenaars system we construct a separation of variables for the classical n particle ruijsenaars system the relativistic analog of the elliptic calogero moser system the separated coordinates appear as the poles of the properly normalised eigenvector baker akhiezer function of the corresponding lax matrix two different normalisations of the ba functions are analysed the canonicity of the separated variables is verified with the use of the r matrix technique the explicit expressions for the generating function of the separating canonical transform are given in the simplest cases n 2 and n 3 taking the nonrelativistic limit we also construct a separation of variables for the elliptic calogero moser system

calogero moser hierarchy and kp hierarchy identification of the hierarchy of calogero moser dynamical systems with the dynamical systems of poles of certain solutions to the kp hierarchy is achieved

spectra of non selfadjoint hill s operators and a class of riemann surfaces i am grateful to a boutet de monvel for the invitation to laboratory of mathematical physics and geometry university paris 7 where the initial version of this paper was written it gives me a special pleasure to thank h mckean for his permanent interest and support of my study of nonselfadjoint periodic operators i am thankful to the referees of this paper whose criticism helped me to improve the exposition the final version of the paper was written at the ben gurion university of the negev the financial support of the ministry of science of israel is gratefully appreciated

singularities of the complex korteweg de vries flows on etudie le probleme isospectral de l equation de hill complexe etant donne 2g 1 valeurs propres simples dans le plan complexe trouver la classe spectrale correspondante q on considere la caracterisation de la classe isospectrale et ses sections singulieres on construit q a l aide des flots non lineaires de korteweg de vries dans c 1 les fonctions regulieres de periode 1

resolvent growth and birkhoff regularity abstract we prove a long standing conjecture in the theory of two point boundary value problems that unconditional basisness implies birkhoff regularity it is a corollary of our two main results minimal resolvent growth along a sequence of points implies nonvanishing of a regularity determinant and sparseness of nth order roots of eigenvalues in small sectors provided that eigen and associated functions of the boundary value problem form an unconditional basis considerations are based on a new direct method exploiting almost orthogonality of birkhoff s solutions of the equation l y y this property was discovered earlier by the author

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

characterization of the potential smoothness of one dimensional dirac operator subject to general boundary conditions and its riesz basis property abstract the one dimensional dirac operator with periodic potential v 0 p x q x 0 where p q l 2 0 subject to periodic antiperiodic or a general strictly regular boundary condition bc has discrete spectrums it is known that for large enough n in the disk centered at n of radius 1 2 the operator has exactly two periodic if n is even or antiperiodic if n is odd eigenvalues n and n counted according to multiplicity and one eigenvalue n b c corresponding to the boundary condition b c we prove that the smoothness of the potential could be characterized by the decay rate of the sequence n b c n where n b c n b c n and n n n furthermore it is shown that the dirac operator with periodic or antiperiodic boundary condition has the riesz basis property if and only if sup n 0 n b c n is finite

spectral parametrization of non selfadjoint hill s operators abstract for the set of hill operators l d 2 dx 2 q x with periodic potentials q x l 2 0 a spectral parametrization is given using a complete system of independent parameters

generalized lame operators we introduce a class of multidimensional schrodinger operators with elliptic potential which generalize the classical lame operator to higher dimensions one natural example is the calogero moser operator others are related to the root systems and their deformations we conjecture that these operators are algebraically integrable which is a proper generalization of the finite gap property of the lame operator using earlier results of braverman etingof and gaitsgory we prove this under additional assumption of the usual liouville integrability in particular this proves the chalykh veselov conjecture for the elliptic calogero moser problem for all root systems we also establish algebraic integrability in all known two dimensional cases a general procedure for calculating the bloch eigenfunctions is explained it is worked out in detail for two specific examples one is related to the b2 case another one is a certain deformation of the a2 case in these two cases we also obtain similar results for the discrete versions of these problems related to the difference operators of macdonald ruijsenaars type

characterization of potential smoothness and the riesz basis property of the hill schrodinger operator in terms of periodic antiperiodic and neumann spectra abstract the hill operators l y y v x y considered with complex valued periodic potentials v and subject to periodic antiperiodic or neumann boundary conditions have discrete spectra for sufficiently large n close to n 2 there are two periodic if n is even or antiperiodic if n is odd eigenvalues n n and one neumann eigenvalue n we study the geometry of the spectral triangle with vertices n n n and show that the rate of decay of triangle size characterizes the potential smoothness moreover it is proved for v l p 0 p 1 that the set of periodic antiperiodic root functions contains a riesz basis if and only if for even respectively odd n sup n n n n n n

dependence of eigenvalues of sturm liouville problems on the boundary abstract the eigenvalues of sturm liouville sl problems depend not only continuously but smoothly on boundary points the derivative of the n th eigenvalue as a function of an endpoint satisfies a first order differential equation this for arbitrary separated or coupled self adjoint regular boundary conditions in addition as the length of the interval shrinks to zero all higher eigenvalues march off to plus infinity this is also true for the first i e lowest dirichlet eigenvalue but not for the lowest neumann eigenvalue the latter has a finite limit

treibich verdier potentials and the stationary m kdv hierarchy here x x col co3 denotes the elliptic weierstrass function with fundamental periods f p 2a l 2co3 im co3 wl 5 0 see 1 ch 18 in the special case where col is real and co3 is purely imaginary the potential q is real valued and ince s striking result 31 in modern spectral theoretic terminology yields the fact that the self adjoint operator l associated with the differential expression d 2 q in l2 ir has a finite gap spectrum of the type