on isospectral periodic potentials in rn ii on considere un reseau a n dimensions l dans r n et un potentiel q satisfaisant q x d q x pour d l on considere spec k q l ensemble des avec multiplicites telles que u qu u a une solution satisfaisant u x d u x exp 2i k d d l

relative symmetric polynomials in this article we introduce the notion of a relative symmetric polynomial with respect to a permutation group and an irreducible character and we give answers for some natural questions about their structures in order to study symmetric polynomials with respect to linear characters we define the concept of relative vandermonde polynomial finally we present some interesting research problems concerning relative symmetric polynomials

on the period spectrum of a symplectic mapping let m be a compact riemannian manifold it has been known for a long time that the singularities of the wave trace trace cos t are located at the periods of the closed geodesics do these singularities also contain information about the geometry of m in the neighborhood of a closed geodesic we prove that the birkhoff canonical form of the poincare map can be determined from the singularities of the wave trace

spectral gaps for periodic schrodinger operators with strong magnetic fields we consider schrodinger operators h h ih d a ih d a with the periodic magnetic field b da on covering spaces of compact manifolds using methods of a paper by kordyukov mathai and shubin 14 we prove that under some assumptions on b there are in arbitrarily large number of gaps in the spectrum of these operators in the semiclassical limit of the strong magnetic field h 0

spectral properties of the periodic magnetic schrodinger operator in the high energy region two dimensional case the goal is to investigate spectral properties of the operator h i a x 2 a0 x in the two dimensional situation a x a0 x being periodic we construct asymptotic formulae for bloch eigenvalues and eigenfunctions in the high energy region describe properties of isoenergetic curves in the space of quasimomenta and give a new proof of the bethe sommerfeld conjecture

semiclassical asymptotics gauge fields and quantum chaos a gauge field is given by a connection on a principal bundle p m we consider the semiclassical behavior of a family of schrodinger operators associated with a gauge field in the limit as h 0 we relate the spectral theory of such operators to behavior of the hamiltonian flow on the natural phase space associated to a gauge field examining in particular situations where this flow exhibits chaotic behavior

equality of symmetrized tensors and the flag variety author s berget andrew abstract in this note we give a transparent proof of a result of da cruz and dias da silva on the equality of symmetrized decomposable tensors this will be done by explaining that their result follows from the fact that the coordinate ring of a flag variety is a unique factorization domain

band asymptotics in two dimensions let x be a rank one symmetric space e g s rp cp and let a be its laplace beltrami operator it is known that the spectrum of a is discrete and consists of a sequence of eigenvalues a k k a k 0 1 2 a 1 1 each occurring with multiplicity n b n l k o k 1 2 n being the dimension of x and a and b being positive integers which depend only on x let q be a zeroth order self adjoint pseudodifferential operator if we perturb a by adding q to it the eigenvalues 1 1 break up into bands

on isospectral periodic potentials on a discrete lattice i on considere une version discretisee et generalisee de l equation de schrodinger et on etudie le probleme aux valeurs propres

small ħ asymptotics for quantum partition functions associated to particles in external yang mills potentials to a gauge field on a principalg bundlep m is associated a sequence of quantum mechanical hamiltonians as planck s constant 0 and a sequence of representations n ofg is taken this paper studies the associated quantum partition functions trace exp th n and produces a complete asymptotic expansion as 0 1 n of which the principal term proportional to the classical partition function is the familiar classical limit

flags and equality of tensors abstract we state a necessary and sufficient condition for equality of decomposable symmetrized tensors when the symmetrizer is associated to an irreducible character of s p corresponding to the partition of p and the tensors are the images by this symmetrizer of the tensor product of a family of vectors of rank partition we make an application of this result to the representation of flags by projective points

on the asymptotic decay of l2 solutions of one body schrodinger equations in unbounded domains the asymptotic decay of l 2 solutions of schrodinger equations v 0 in r x er n x r r is investigated where v x v 1 r v 2 x with v 1 for r and with some e 0 for large r under additional assumptions on the decay of v 1 pointwise upper bounds to and lower bounds to the spherical average of are given showing the same asymptotics for r for the case v const 0 for r investigated in 8 a simplified treatment is given

homogeneous quantization and multiplicities of group representations let b be a compact manifold a cone over b is a principal r bundle x with base b let a x a x be the mapping associated with the action of a r on x x is called a symplectic cone if it possesses a symplectic form such that a a a compact lie group g is said to act in a homogeneous fashion on x if it acts on x in such a way that both and the principal bundle structure are preserved it is known that to such an action one can associate in a fairly canonical way a representation of g on a hilbert space h see 3 in this paper we propose a symplectic recipe for the multiplicities with which h decomposes into g irreducibles and show that this recipe is correct generically

on the constructive determination of spectral invariants of the periodic schrodinger operator with smooth potentials in this paper we develop the asymptotic formulae obtained in my previous papers for the band functions and the bloch functions of the schrodinger operator with the smooth periodic potentials then using these formulae we determine constructively a family of spectral invariants of this operator from the given band functions some of these invariants generalize the well known invariants and others are entirely new the new invariants are explicitly expressed by fourier coefficients of the potential which present the possibility of determining the potential constructively by using the band functions as input data

spectral theory for periodic schrodinger operators with reflection symmetries let h v be defined on d with smooth potential v such that v x v x n rm for all n in mathbb z d in addition we assume that v t jx v x j 1 dots d where t j x 1 dots x j 1 x j x j 1 dots x d x 1 dots x j 1 x j x j 1 dots x d this is a periodic schrodinger operator with additional reflection symmetries we investigate the associated floquet operators h q q 0 1 d in particular we show that the associated lowest eigenvalues q are simple if q q 1 q 2 q d satisfies q j 1 2 for each j 1 2 d

exponential lower bounds to solutions of the schrodinger equation lower bounds for the spherical average for a large class of generalizedn body schrodinger operators h we show that ife inf ess h and is an eigenfunction ofh with eigenvaluee then begin array 20 c lim r to infty end array r 1 ln left int limits s n 1 psi r omega 2 d omega right 1 2 alpha 0 with 0 2 e a threshold similar results are given fore

absence of singular continuous spectrum for certain self adjoint operators we give a sufficient condition for a self adjoint operator to have the following properties in a neighborhood of a pointe of its spectrum a its point spectrum is finite b its singular continuous spectrum is empty c its resolvent satisfies a class of a priori estimates

spectral asymptotics of periodic elliptic operators we demonstrate that the structure of complex second order strongly elliptic operators h on bf r d with coefficients invariant under translation by bf z d can be analyzed through decomposition in terms of versions h z z in bf t d of h with z periodic boundary conditions acting on l 2 bf i d where bf i 0 1 rangle if the s emigroup s generated by h has a holder continuous integral kernel satisfying gaussian bounds then the semigroups s z generated by the h z have kernels with similar properties and z mapsto s z extends to a function on bf c d backslash 0 which is analytic with respect to the trace norm the sequence of semigroups s m z obtained by rescaling the coefficients of h z by c x to c mx converges in trace norm to the semigroup widehat s z generated by the homogenization widehat h z of h z these convergence properties allow asymptotic analysis of the spectrum of h

band asymptotics with non smooth potentials let s denote the n dimensional sphere with its canonical riemannian metric and a its laplace beltrami operator if q is a real valued function on s we can consider the self adjoint operator s a q for the case when q is smooth weinstein proved in w the following very beautiful theorem on the asymptotic behavior of the spectrum of s let dk k k n 1 be the kth eigenvalue of a and let d be its multiplicity by the minimax principle the eigenvalues of s can be written in the form ak p i l dk where ipclk il is bounded by a constant independent of k and i let 0 denote the space of oriented closed geodesics on s and let dp be the unique probability measure on 0 which is rotationally invariant by integrating q along closed geodesics we obtain a smooth function s on 0 which we call the x ray transform of q then weinstein s theorem is

l 2 exponential lower bounds to solutions of the schrodinger equation we study decay properties of solutions of the schrodinger equation v e typical of our results is one which shows that ifv o x 1 2 at infinity or ifv is a homogeneousn body potential for example atomic or molecular then ife 0 and alpha sqrt e e alpha left x right psi notin l 2 left mathbb r n right we also construct examples to show that previous essential spectrum dependent upper bounds can be far from optimal if is not the ground state

semiclassical spectra of gauge fields abstract we study the asymptotic behavior of the eigenvalues of the schrodinger operator with a vector potential on a compact manifold as planck s constant tends to zero we obtain estimates in terms of periodic trajectories of wong s flow which are uniform in the charge parameter

inverse spectral results on two dimensional tori is much more complicated than the corresponding problem on surfaces of genus 2 on one hand for certain classes of potentials one can reconstruct v from the spectrum of 1 1 on the other hand there are potentials that have lots of isospectral deformations in their analysis of 1 1 an important element is the decomposition of the fourier series of v into primitive summands and we begin here by saying a few words about this decomposition since it will also be important in the results to be described an element of a lattice is said to be primitive if it is not a positive integer multiple of another element of the lattice now let a e27i w x ct e r

on the constructive determination of the periodic potentials from the bloch eigenvalues in this paper we consider the three dimensional schrodinger operator with a periodic relative to a lattice of potential q we construct a set d of trigonometric polynomials such that d is dense in where s 3 in the topology any element q of the set d can be determined constructively and uniquely modulo inversion and translation q x q x q x q x where from the given bloch eigenvalues of the schrodinger operator with the potential q

perturbation theory for the decay rate of eigenfunctions in the generalized n body problem simple examples are known where eigenfunctions decay faster than the usual upper bounds would lead one to believe we develop aspects of the perturbation theory of the decay rate of eigenfunctions as measured by radial exponential weights we show that generically in a baire category sense eigenfunction decay rates are governed by the lowest threshold

on the structure of spectra of periodic elliptic operators the paper discusses the problem of absolute continuity of spectra of periodic elliptic operators a new result on absolute continuity for a matrix operator of schrodinger type is obtained it is shown that all types of operators for which the absolute continuity has previously been established can be reduced to this one it is also discovered that some natural generalizations stumble upon an obstacle in the form of non triviality of a certain analytic bundle on the two dimensional torus

new conditions for equality of decomposable symmetrized tensors we determine conditions for equality of decomposable symmetrized tensors in arbitrary symmetry classes based on these conditions we derive results for equality of decomposable symmetrized tensors associated to certain special group characters

on the spectrum of a class of second order periodic elliptic differential operators under an additional symmetry condition we prove that the spectrum of a second order self adjoint elliptic differential operator with periodic coefficients is purely absolutely continuous

equality of immanantal decomposable tensors we state a necessary and sufficient condition for equality of two nonzero decomposable symmetrized tensors when the symmetrizer is associated with an irreducible character of the symmetric group of degree m

absolute continuity in periodic waveguides we study second order elliptic operators with periodic coefficients in two dimensional simply connected periodic waveguides with the dirichlet or neumann boundary conditions it is proved that under some mild smoothness restrictions on the coefficients such operators have purely absolutely continuous spectra the proof follows a method suggested previously by a morame to tackle periodic operators with variable coefficients in dimension 2 2000 mathematical subject classification 35j10 35p05 35j25

inverse spectral problem for the schrodinger equation with periodic vector potential for the schrodinger operator with periodic magnetic vector and electric scalar potentials a new system of spectral invariants is found these invariants are enough to prove the rigidity of isospectral deformations in the class of generic even and real analytic magnetic and electric potentials

noncommutative bloch theory for differential operators which are invariant under the action of an abelian group bloch theory is the preferred tool to analyze spectral properties by shedding some new noncommutative light on this we motivate the introduction of a noncommutative bloch theory for elliptic operators on hilbert c modules it relates properties of c algebras to spectral properties of module operators such as band structure weak genericity of cantor spectra and absence of discrete spectrum it applies e g to differential operators invariant under a projective group action such as schrodinger dirac and pauli operators with periodic magnetic field as well as to discrete models such as the almost matthieu equation and the quantum pendulum

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

on positive solutions of second order elliptic equations stability results and classification soit p un operateur uniformement elliptique d ordre 2 defini dans un domaine cr n soit p le cone convexe de toutes les solutions positives de l equation pu 0 dans on etudie les proprietes generales de p pour p et generaux

an overview of periodic elliptic operators the article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas close attention is paid to studying analytic properties of bloch and fermi varieties which influence significantly most properties of such operators the approaches described are applicable not only to the standard model example of schrodinger operator with periodic electric potential v x but to a wide variety of elliptic periodic equations and systems equations on graphs operator and other operators on abelian coverings of compact bases many important applications are mentioned however due to the size restrictions they are not dealt with in details

exponential bounds and absence of positive eigenvalues for n body schrodinger operators for a large class ofn body potentialsv we prove that if is an eigenfunction of v with eigenvaluee then sup 2 e 0 exp x l2 is either a threshold or consequences of this result are the absence of positive eigenvalues and optimal l2 exponential lower bounds

spectral analysis of n body schrodinger operators for a large class of two body potentials we solve two of the main problems in the spectral analysis of multiparticle quantum hamiltonians explicitly we prove that the point spectrum lies in a closed countable set and describe that set in terms of the eigenvalues of hamiltonians of subsystems and that there is no singular continuous spectrum we accomplish this by extending mourre s work on three body problems to n body problems

perturbation of embedded eigenvalues in the generalized n body problem we discuss the perturbation of continuum eigenvalues without analyticity assumptions among our results we show that generally a small perturbation removes these eigenvalues in accordance with fermi s golden rule thus generically in a baire category sense the schrodinger operator has no embedded non threshold eigenvalues

on the kernel of the derivation operator abstract we state a necessary and sufficient condition for a symmetrized tensor to belong to the kernel of the derivation operator the derivation operator a generalization of the associated transformation introduced by m marcus in o shisha ed inequalities ii academic press new york 1970 p 223 is a linear operator on the range of the symmetrizer when the symmetrizer is associated to an absolutely irreducible character of s m corresponding to the partition of m and the tensor is the image by this symmetrizer of the tensor product of a family of vectors of rank partition

the absolute continuity of the spectrum of maxwell operator in a periodic media we prove that the spectrum of maxwell operator b e rot e a x rot b x b with non negative and periodic functions a x and b x is absolutely continuous

classical limits for quantum particles in external yang mills potentials we consider the limit 0 for nonrelativistic quantum particles moving in external yang mills potentials it is shown that the partition function and the solutions of the equations of motion converge to their corresponding classical counterparts

floquet bundles for scalar parabolic equations for linear scalar parabolic equations such as u t u xx a t x u x b t x u on a finite interval 0 x with various boundary conditions we obtain canonical floquet solutions u n t x these solutions are characterized by the property that z u n t x n for all te where z denotes the zero crossing lap number of matano the coefficients a t x and b t x are not assumed to be periodic in t but if they are the solutions u n t x reduce to the standard floquet solutions our results may naturally be expressed in the language of linear skew product flows in this context we obtain for each n 1 an exponential dichotomy between the bundles span u n 1 and overline span u n cdot cdot n n 1 infty

colorings and equality of tensors we study necessary conditions for equality of two nonzero star products of vectors in symmetry classes of tensors associated to irreducible characters of the symmetric group of degree m these conditions depend upon the notion of coloring of a family of vectors introduced in linear and multilinear algebra 27 1990 25

the multilinear algebra of jose dias da silva and the portuguese school of mathematics abstract in portugal multilinear algebra has a strong group of researchers that achieved in the last years some impressive results in this area this paper surveys the results obtained by this group and the current state of knowledge on the different problems

an algorithm for finding the periodic potential of the three dimensional schrodinger operator from spectral invariants in this paper we investigate the three dimensional schr dinger operator with a periodic relative to a lattice of potential q a special class v of the periodic potentials is constructed which is easily and constructively determined from the spectral invariants first we give an algorithm for the unique determination of the potential q v of the three dimensional schr dinger operator from the spectral invariants that were determined constructively from the given bloch eigenvalues then we consider the stability of the algorithm with respect to the spectral invariants and bloch eigenvalues finally we prove that there are no other periodic potentials in the set of large class of functions whose bloch eigenvalues coincides with the bloch eigenvalues of q v

continuity and nodal properties near infinity for solutions of 2 dimensional schrodinger equations on considere des solutions w 2 2 a valeurs reelles x de v e 0 pour x r r x r 2 x r r r 0 on suppose que e 0

on a quantum chaos theorem of r schrader and m taylor abstract we extend results of r schrader and m taylor on the semi classical asymptotics of eigenfunctions of a quantum hamiltonian for a particle in a gauge field our method is based on the guillemin sternberg uribe theory of fuzzy ladders and reductions of hamiltonian systems with symmetry

splitting of the low landau levels into a set of positive lebesgue measure under small periodic perturbations we study the spectral properties of a two dimensional schrodinger operator with a uniform magnetic field and a small external periodic field where and are small parameters representing as the direct integral of one dimensional quasi periodic difference operators with long range potential and employing recent results of e i dinaburg about anderson localization for such operators we assume to be typical irrational we construct the full set of generalised eigenfunctions for the low landau bands we also show that the lebesgue measure of the low bands is positive and proportional in the main order to

circular symmetry and the trace formula let m be a compact n dimensional manifold without boundary and let p be a positive first order elliptic self adjoint classical pseudodifferential operator on lz m the hilbert space of half densities on m the spectrum of p is discrete and accumulates only at we denote the eigenvalues with multiplicities by 2o 21 2 recall that the singularities of the inverse fourier transform of the spectral measure