chow s theorem and universal holonomic quantum computation a theorem from control theory relating the lie algebra generated by vector fields on a manifold to the controllability of the dynamical system is shown to apply to holonomic quantum computation conditions for deriving the holonomy algebra are presented by taking covariant derivatives of the curvature associated with a non abelian gauge connection when applied to the optical holonomic computer these conditions determine that the holonomy group of the two qubit interaction model contains su 2 su 2 in particular a universal two qubit logic gate is attainable for this model

spectral properties of non selfadjoint sturm liouville operator with operator coefficient abstract in this paper we consider the sturm liouville operator l generated in l 2 r h by the differential expression l y y q x y 0 x with the boundary condition y 0 0 where q x is a non selfadjoint completely continuous operator in h for every x 0 here h is a separable hilbert space b h denotes the space of bounded operators in h and l 2 r h denotes the space of square integrable strongly measurable vector valued functions defined on 0 in particular we find some special solutions of the equation l y 2 y including jost solution then investigate the point spectrum of l under certain conditions on q x we obtain the resolvent of l if q x is quasi selfadjoint i e there exists p b h such that p 1 b h p is positive and q x p q x p 1 for every x 0 we also show that l has a finite number of spectral singularities

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

on the finiteness of differential invariants abstract in this paper we show how weil s theory of near points yields a new light on the classical approaches to the study of the differential invariants of a sheaf of tangent vector fields we give conditions for the existence of invariant derivations for a sheaf of tangent vector fields which allows to apply lie s algorithm to obtain new differential invariants as quotients of jacobian determinants of known ones we give sufficient conditions for the asymptotic stability of the symbol of a sheaf of tangent vector fields and prove our main result a finiteness theorem for the differential invariants of a sheaf of lie algebras which simplifies and improves on the treatment given in j differential geom 10 1975 249 416

on isospectral sets of jacobi operators we consider the inverse spectral problem for a class of reflectionless bounded jacobi operators with empty singularly continuous spectra our spectral hypotheses admit countably many accumulation points in the set of eigenvalues as well as in the set of boundary points of intervals of absolutely continuous spectrum the corresponding isospectral set of jacobi operators is explicitly determined in terms of dirichlet type data

on the universality of almost every quantum logic gate lloyd phys rev lett 75 346 1995 showed that almost every quantum logic gate is universal in the sense that it can be used to approximate any unitary transformation the argument relied on a more general fact whose proof was not given in detail we give a complete proof of this more general fact

a proof of the local borg marchenko theorem in 2 5 a local version of a basic uniqueness theorem in inverse spectral theory is given this paper gives a simple proof of this theorem

symmetries of pde systems and correspondences between jet spaces abstract let m be a manifold a pde system r j m 1 m can be prolonged to another one r t m jimenez et al 2005 10 in analogy with the higher order symmetries symmetries of r will be called higher dimensional symmetries of r for a broad class of pde systems we prove that every infinitesimal or finite symmetry of r comes from another one of r we show that r does not have internal infinitesimal symmetries modulo trivial symmetries this fact allows us in the infinitesimal case to compute the internal symmetries of r as external symmetries of r we also give an algorithmic method to obtain solutions of r invariant by a given internal symmetry

time optimal control of finite quantum systems we investigate time optimal control of finite quantum systems in the born approximation a bang bang principle is found to follow from a result in c a akemann and j anderson mem amer math soc 458 1991 we also prove existence of time optimal controls characterize when they are unique and assuming uniqueness explicitly describe them

borg type theorems for matrix valued schrodinger operators abstract a borg type uniqueness theorem for matrix valued schrodinger operators is proved more precisely assuming a reflectionless potential matrix and its spectrum a half line 0 we derive the triviality of the potential matrix our approach is based on trace formulas and matrix valued herglotz representation theorems as a by product of our techniques we obtain an extension of borg s classical result from the class of periodic scalar potentials to the class of reflectionless matrix valued potentials

a remark on goldschmidt s theorem on formal integrability abstract in this paper we apply weil bundle techniques to the study of formal integrability for systems of nonlinear partial differential equations we clarify the role of the curvature and show that both the statement and proof of goldschmidt s criterion on formal integrability are of algebraic nature this fact allows us to obtain a version of this theorem which holds for smooth algebraic or analytic manifolds finally we give a definition for the characteristic co vectors of a system of partial differential equations and how their relationship with the cauchy kowalevski normal form

non selfadjoint matrix sturm liouville operators with spectral singularities in this paper we investigate discrete spectrum of the non selfadjoint matrix sturm liouville operator l generated in l 2 r s by the differential expression y y q x y x r 0 and the boundary condition y 0 0 under the conditionsupx r exp ex q x 0using the uniqueness theorem of analytic functions we prove that l has a finite number of eigenvalues and spectral singularities with finite multiplicities

almost any quantum logic gate is universal it is shown that if one can apply some hamiltonian repeatedly to a few variables at a time one can in general effect any desired unitary time evolution on an arbitrarily large number of variables as a result almost any quantum logic gate with two or more inputs is computationally universal in that copies of the gate can be wired together to effect any desired logic circuit and to perform any desired unitary transformation on a set of quantum variables

quantum computational networks the theory of quantum computational networks is the quantum generalization of the theory of logic circuits used in classical computing machines quantum gates are the generalization of classical logic gates a single type of gate the universal quantum gate together with quantum unit wires is adequate for constructing networks with any possible quantum computational property

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

an inverse problem for the matrix schrodinger equation abstract by modifying and generalizing some old techniques of n levinson a uniqueness theorem is established for an inverse problem related to periodic and sturm liouville boundary value problems for the matrix schrodinger equation

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

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 local borg marchenko uniqueness results we provide a new short proof of the following fact first proved by one of us in 1998 if two weyl titchmarsh m functions m j z of two schrodinger operators h j d 2 dx 2 q j j 1 2 in l 2 o r o r are exponentially close that is m 1 z m 2 z z o e 2 im z 1 2 a o r then q 1 q 2 a e on o the result applies to any boundary conditions at x o and x r and should be considered a local version of the celebrated borg marchenko uniqueness result which is quickly recovered as a corollary to our proof moreover we extend the local uniqueness result to matrix valued schrodinger operators

an eigenfunction expansion for a quadratic pencil of a schrodinger operator with spectral singularities in this paper we consider the operatorlgenerated inl2 r by the differential expressionl y y q x 2 p x 2 y x r 0 and the boundary conditiony 0 0 wherepandqare complex valued functions andpis continuously differentiable onr we derive a two fold spectral expansion ofl in the sense of keldysh 1951 soviet math dokl 77 11 14 1971 russian math survey26 15 44 engl transl in terms of the principal functions under the conditionslimx p x 0 supx r eex q x p x 0 taking into account the spectral singularities also we investigate the convergence of the spectral expansion

invariant parameterization and turbulence modeling on the beta plane abstract invariant parameterization schemes for the eddy vorticity flux in the barotropic vorticity equation on the beta plane are constructed and then applied to turbulence modeling this construction is realized by the exhaustive description of differential invariants for the maximal lie invariance pseudogroup of this equation using the method of moving frames which includes finding functional bases of differential invariants of arbitrary order a minimal generating set of differential invariants and a basis of operators of invariant differentiation in an explicit form special attention is paid to the problem of two dimensional turbulence on the beta plane it is shown that classical hyperdiffusion as used to initiate the energy enstrophy cascades violates the symmetries of the vorticity equation invariant but nonlinear hyperdiffusion like terms of new types are introduced and then used in the course of numerically integrating the vorticity equation and carrying out freely decaying turbulence tests it is found that the invariant hyperdiffusion scheme is closely reproducing the theoretically predicted k 1 shape of enstrophy spectrum in the enstrophy inertial range by presenting conservative invariant hyperdiffusion terms we also demonstrate that the concepts of invariant and conservative parameterizations are consistent

symmetry transformations of an ideal steady fluid flow determined by a potential function first we consider a transformation of 3d trajectories fluid particle paths of inviscid steady flows using the dual stream function approach for the local representation of velocity fields u x y z this enables to derive the equation governing the deformation of trajectories by the gradient field along the surface x y z 0 in fact is a symmetry transformation and it looks formally like the filament motion which preserves the curvature then we investigate in detail a fine structure of a lie algebra associated with an extension of the transformation which creates a visual appearance of sliding stream surfaces x y z 0 along itself the minimal set of generating differential invariants is found this set consists of a single invariant which coincides with a hamiltonian function

spectrum and spectral singularities of a quadratic pencil of a schrodinger operator with a general boundary condition abstract in this article we investigate the spectrum and the spectral singularities of the quadratic pencil of schrodinger operatorlgenerated inl2 r by the differential expression l y y q x 2 p x 2 y x r 0 and the boundary condition 0 k x y x dx y 0 y 0 0 wherep q and k are complex valued functions p is continuously differentiable onr k l2 r and c with 0 discussing the spectrum we prove that l has a finite number of eigenvalues and spectral singularities with finite multiplicities if the conditions lim x p x 0 sup x r e e x q x p x k x 0 later we investigate the properties of the principal functions corresponding to the spectral singularities moreover some results about the spectrum oflare applied to non selfadjoint sturm liouville and klein gordons wave operators

trace formulas and borg type theorems for matrix valued jacobi and dirac finite difference operators abstract borg type uniqueness theorems for matrix valued jacobi operators h and supersymmetric dirac difference operators d are proved more precisely assuming reflectionless matrix coefficients a b in the self adjoint jacobi operator h as a s b with s the right left shift operators on the lattice z and the spectrum of h to be a compact interval e e e e we prove that a and b are certain multiples of the identity matrix an analogous result which however displays a certain novel nonuniqueness feature is proved for supersymmetric self adjoint dirac difference operators d with spectrum given by e 1 2 e 1 2 e 1 2 e 1 2 0 e e our approach is based on trace formulas and matrix valued exponential herglotz representation theorems as a by product of our techniques we obtain the extension of flaschka s borg type result for periodic scalar jacobi operators to the class of reflectionless matrix valued jacobi operators

universality in quantum computation we show that in quantum computation almost every gate that operates on two or more bits is a universal gate we discuss various physical considerations bearing on the proper definition of universality for computational components such as logic gates

hamiltonian systems of limit point or limit circle type with both endpoints singular abstract a linear hamiltonian system j y a b y is considered on an open interval a b where both a and b are singular the system is assumed to be of limit point or limit circle type at the endpoints a theory of boundary problems for such systems is developed explicit boundary conditions are given resolvent operators constructed and unique solutions established the results given extend to hamiltonian systems a theory of singular boundary value problems due to m h stone and k kodaira

integrability of lie equations and pseudogroups in this paper the theory of jets based on weil s near points is applied to lie equations and pseudogroups linear systems of partial differential equations are interpreted in a canonical way as distributions on the fibre bundles of invertible jets invariant under translations we prove the two fundamental theorems for lie equations and generalize the results of rodrigues a geometric correspondence between linear and nonlinear lie equations is given and the symbols of a linear lie equation and its prolongations are canonically identified with the symbols of their attached nonlinear equations from this fact we deduce that a linear lie equation verifies the conditions of goldsmichmidt s criterion on formal integrability if and only if its attached nonlinear lie equation satisfies them locally finally we define the cartan 1 form on the fibre bundle of invertible jets and give a global form to the equivalence between the lie and cartan definitions of continuous groups

internal external and generalized symmetries backlund s theorem which characterizes contact transformations is generalized to give an analogous characterization of internal symmetries of systems of differential equations for a wide class of systems of differential equations we prove that every internal symmetry comes from a first order generalized symmetry and conversely every first order generalized symmetry satisfying certain explicit contact conditions determines an internal symmetry we analyze the contact conditions in detail deducing powerful necessary conditions for a system of differential equations that admit genuine internal symmetries i e ones which do not come from classical external symmetries applications include a direct proof that both the internal symmetry group and the first order generalized symmetries of a remarkable differential equation due to hilbert and cartan are the noncompact real form of the exceptional simple lie group g2

symmetry reduction of ordinary finite difference equations using moving frames the technique of equivariant moving frames is incorporated into the classical symmetry reduction method of ordinary finite difference equations using the recurrence relations for the finite difference invariants computations are performed symbolically without relying on the coordinate expressions of the canonical variables and the difference invariants

differential invariant algebras of lie pseudo groups abstract the aim of this paper is to describe in as much detail as possible and constructively the structure of the algebra of differential invariants of a lie pseudo group acting on the submanifolds of an analytic manifold under the assumption of local freeness of a suitably high order prolongation of the pseudo group action we develop computational algorithms for locating a finite generating set of differential invariants a complete system of recurrence relations for the differentiated invariants and a finite system of generating differential syzygies among the generating differential invariants in particular if the pseudo group acts transitively on the base manifold then the algebra of differential invariants is shown to form a rational differential algebra with non commuting derivations the essential features of the differential invariant algebra are prescribed by a pair of commutative algebraic modules the usual symbol module associated with the infinitesimal determining system of the pseudo group and a new prolonged symbol module constructed from the symbols of the annihilators of the prolonged pseudo group generators modulo low order complications the generating differential invariants and differential syzygies are in one to one correspondence with the algebraic generators and syzygies of an invariantized version of the prolonged symbol module our algorithms and proofs are all constructive and rely on combining the moving frame approach developed in earlier papers with grobner basis algorithms from commutative algebra

a transformation of pde systems related to the drach theorem we will define a new transformation of pde systems as follows given a particular pde system f there is a new system f whose solutions are the spaces of elements attached to the solutions of f we will show that the system f is a second order pde system in a single unknown as an application we will derive as well a global version of the drach theorem

maurer cartan equations for lie symmetry pseudogroups of differential equations a new method of constructing structure equations of lie symmetry pseudogroups of differential equations dispensing with explicit solutions of the infinitesimal determining systems of the pseudogroups is presented and illustrated by the examples of the kadomtsev petviashvili and korteweg de vries equations

principal functions of non selfadjoint matrix sturm liouville equations in this paper we investigate the principal functions corresponding to the eigenvalues and the spectral singularities of the boundary value problem y q x y l 2y x r 0 y 0 0 where q is a non selfadjoint matrix valued function

invariant submanifold flows given a lie group acting on a manifold our aim is to analyze the evolution of differential invariants under invariant submanifold flows the constructions are based on the equivariant method of moving frames and the induced invariant variational bicomplex applications to integrable soliton dynamics and to the evolution of differential invariant signatures used in equivalence problems and object recognition and symmetry detection in images are discussed

the trace formula for schrodinger operators on the line this paper discusses certain aspects of the spectral and inverse spectral problems for the schrodinger operator l q frac d 2 dx 2 q x for q x c the space of bounded continuous functions the trace formula of the title is the relation q 0 lambda 0 sum limits j 1 infty lambda 2j lambda 2j 1 2 mu j with appropriate choices of j j 0 and j j 1 which is a familiar relation in the theory of hill s equation we characterize the set subseteq c of potentials for which this holds further extensions of the theory of hill s equation are also obtained from the spectrum l q a torust q is constructed which is in general infinite dimensjional everyq x can be mapped to a continuous path ont q described by the auxiliary spectrum j j 1 under certain geometrical conditions on l q this path is the orbit of ac 1 vector field ont q and the mapping extends to one from the hull q overline q x xi xi in mathbb r to the closure of this orbit in particular q is compact these results have applications in the theory of schrodinger operators with ergodic potentials

absolute summability of the trace relation for certain schrodinger operators a recently established general trace formula for one dimensional schrodinger operators is systematically studied in the context of short range potentials potentials which approach different spatial asymptotes sufficiently fast and appropriate impurity defect interactions in one dimensional solids we prove the absolute summability of the trace formula and establish its connections with scattering quantities such as reflection coefficients in each case

almost periodicity of some random potentials in this paper we study some random potentials with purely absolutely continuous spectrum in one dimension there are no examples of random potentials with absolutely continuous spectrum that are not almost periodic though it is known from kotani 3 that such potentials are deterministic hence one could conjecture that any random potential with purely absolutely continuous spectrum is almost periodic we show this result for random potentials with infinitely many bands in their spectrum such that the gaps behave well in the finite bands case concini and johnson l discussed the same problem a precise definition of the behavior of the gaps could be found in condition 2 3 we make use of the results of levitan 7 kotani 3 4 and moser 9 for doing this we also discuss the behavior of the lyapunov exponent asymptotically and its relation to the smoothness of the potentials

almost periodic schrodinger operators iii the absolutely continuous spectrum in one dimension we studyh d2 dx2 v x withv x limit periodic e g v x ancos x 2n with an we prove that for a genericv and for generican in the explicit example h is a cantor nowhere dense perfect set for a dense set the spectrum is both cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous

quadratic pencil of schrodinger operators with spectral singularities discrete spectrum and principal functions abstract in this article we investigated the spectrum of the quadratic pencil of schrodinger operatorsl generated inl2 r by the equation formula and the boundary condition formula whereu vare complex valued functions anduis absolutely continuous in each finite subinterval of r discussing the spectrum we proved thatl has a finite number of eigenvalues and spectral singularities with finite multiplicities if the conditions formula and formula hold it is shown that the principal functions corresponding to eigenvalues ofl are in the spacel2 r and the principal functions corresponding to spectral singularities are in another hilbert space which containsl2 r some results about the spectrum ofl have also been applied to radial klein gordon and one dimensional schrodinger equations

on the arbitrariness of the general solution of an involutive partial differential equation the relationship between the strength of a differential equation as introduced by einstein its cartan characters and its hilbert polynomial is studied using the framework of formal theory previous results are extended to nonlinear equations of arbitrary order and to overdetermined systems the problem of computing the number of arbitrary functions in the general solution is treated finally the effect of gauge symmetries is considered

uniform finite generation of compact lie groups abstract consider a compact connected lie group g and the corresponding lie algebra l let x1 xm be a set of generators for the lie algebra l we prove that g is uniformly finitely generated by x1 xm this means that every element k g can be expressed as k ext1ext2 extl where the indeterminates x are in the set x1 xm t i r i 1 l and the number l is uniformly bounded this extends a previous result by f lowenthal in that we do not require the connected one dimensional lie subgroups corresponding to the x i i 1 m to be compact we link the results to the existence of universal logic gates in quantum computing and discuss the impact on bang bang control algorithms for quantum mechanical systems