phase space tunneling for operators with symbols in a gevrey class phase space tunneling and exponential decay of eigenfunctions in phase space are well known for operators with symbols which are analytic in some neighborhood of the real axis this can be used to prove an adiabatic theorem of exponential order if one assumes the hamiltonian to depend analytically on time however to study compactly supported switching processes one has to weaken the analyticity assumptions here we examine nonanalytic symbols with gevrey class regularity and show that we get an exponential decay of the corresponding eigenfunctions with respect to 1 a as 0 where a 1 the loss of regularity causes a slower decay in the analysis is done using the methods of martinez and its generalization by nakamura an upper bound for the rate of decay is given

the regularized boussinesq equation instability of periodic traveling waves abstract in this work we study the linear instability of periodic traveling waves associated with a generalization of the regularized boussinesq equation by using analytic and asymptotic perturbation theory we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so the linear instability of periodic profiles is obtained with respect to applications of this approach we prove the linear nonlinear instability of cnoidal wave solutions for the modified regularized boussinesq equation and for a system of two coupled boussinesq equations

new criterion of partial asymptotic stability in probability of stochastic differential equations autonomous system of stochastic differential equations dx t f x dt i 1 kg i x dw i t which has a zero solution x 0 is considered it is assumed that there exists a positive definite with respect to part of the variables function v x such that corresponding operator lv is nonpositive it is proved that if the set m 0 x lv 0 x v x 0 does not include entire semitrajectories of the system 2 1 almost surely then the zero solution is asymptotically stable in probability with respect to part of the variables

elementary exponential error estimates for the adiabatic approximation we present an elementary proof that the quantum adiabatic approximation is correct up to exponentially small errors for hamiltonians that depend analytically on the time variable our proof uses optimal truncation of a straightforward asymptotic expansion we estimate the terms of the expansion with standard cauchy estimates

asymptotically fast factorization of integers the paper describes a probabilistic algorithm for finding a factor of any large composite integer n the required input is the integer n together with an auxiliary sequence of random numbers it is proved that the expected number of operations which will be required is o exp 83qn n in in n l 2 for some constant f 0 asymptotically this algorithm is much faster than any previously analyzed algorithm for factoring integers earlier algorithms have all required o na operations where a 1 5

the quantum n body problem this selective review is written as an introduction to the mathematical theory of the schrodinger equation for n particles characteristic for these systems are the cluster properties of the potential in configuration space which are expressed in a simple geometric language the methods developed over the last 40 years to deal with this primary aspect are described by giving full proofs of a number of basic and by now classical results the central theme is the interplay between the spectral theory of n body hamiltonians and the space time and phase space analysis of bound states and scattering states

double wells perturbation series summable to the eigenvalues and directly computable approximations we give a rigorous proof of the analyticity of the eigenvalues of the double well schrodinger operators and of the associated resonances we specialize the rayleigh schrodinger perturbation theory to such problems obtaining an expression for the complex perturbation series uniquely related to the eigenvalues through a summation method by an approximation we obtain new series expansions directly computable still summable which in the case of the herbst simon model can be given in an explicit form

stochastic versions of the lasalle theorem the main aim of this paper is to establish stochastic versions of the well known lasalle stability theorem from these stochastic versions follow many classical results on stochastic stability this shows clearly the power of our new results

reductions among number theoretic problems abstract many number theoretic problems such as integer factorization and the discrete logarithm problem have defied all attempts to classify their complexities thirteen such problems are considered none of which is known either to have a deterministic polynomial time solution or to be complete for any natural complexity class failing this the next best goal is to determine which among these are the easiest and which are the hardest problems toward this end this paper gives an overview of reductions among the problems two reductions are new a deterministic polynomial time reduction from squarefreeness to euler s function n and a probabilistic polynomial time reduction from order modulo a prime power to discrete logarithm modulo a prime power

the magnetic weyl calculus in the presence of a variable magnetic field the weyl pseudodifferential calculus must be modified the usual modification based on the minimal coupling principle at the level of the classical symbols does not lead to gauge invariant formulas if the magnetic field is not constant we present a gauge covariant quantization relying on the magnetic canonical commutation relations the underlying symbolic calculus is a deformation defined in terms of the magnetic flux through triangles of the classical moyal product

comparison theorems for the gap of schrodinger operators on cherche des bornes inferieures aux decompositions de valeur propre en fonction des proprietes geometriques de l ensemble avec v x e ainsi que sur la taille de v sur cet ensemble

general adiabatic evolution with a gap condition we consider the adiabatic regime of two parameters evolution semigroups generated by linear operators that are analytic in time and satisfy the following gap condition for all times the spectrum of the generator consists in finitely many isolated eigenvalues of finite algebraic multiplicity away from the rest of the spectrum the restriction of the generator to the spectral subspace corresponding to the distinguished eigenvalues is not assumed to be diagonalizable

a subexponential time algorithm for computing discrete logarithms over gf p 2 an algorithm for computing discrete logarithms over gf p 2 where p is a prime in subexponential time is described the algorithm is similar to the merkle adleman algorithm for computing logarithms over gf p but it uses quadratic fields as the appropriate algebraic structure it also makes use of the idea of a virtual spanning set due to hellman and reyneri for computing discrete logarithms over gf p m for m growing and p fixed

on eigenfunction decay for two dimensional magnetic schrodinger operators for two dimensional schrodinger operators with a nonzero constant magnetic field perturbed by a magnetic field and a scalar potential both vanishing arbitrarily slow at infinity it is proved that eigenfunctions corresponding to the discrete spectrum decay faster than any exponential under more restrictive conditions on the perturbations even quicker decay is obtained

on the adiabatic theorem for nonself adjoint hamiltonians the authors consider the evolution of a two level system driven by a nonself adjoint hamiltonian h in t and treat the adiabatic limit in to 0 while adiabatic theorem like results do not hold true in general for this case they prove that they are still valid for the subspace corresponding to the eigenvalue having the largest imaginary part least dissipative eigenvalue the theory gives the full asymptotic expansion of the evolution restricted to this subspace the first correction beyond berry s phase is to their best knowledge given explicitly for the first time

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

generation of elements with small modular squares and provably fast integer factoring algorithms finding small modular squares when the modulus is a large composite number of unknown factorization is almost certainly a computationally hard problem this problem arises in a natural way when factoring the modulus by the use of congruences of squares we study here with the help of lattices the set of elements whose squares mod n are small enough less than 0 n213 we obtain a precise description of the gaps between such elements and we develop two polynomial time algorithms that find elements with small modular squares the first is a randomized algorithm that generates such elements in a near uniform way we use it to derive a class of integer factorization algorithms the fastest of which provides the best rigorously established probabilistic complexity bound for integer factorization algorithms the second algorithm is deterministic and often finds amongst the neighbors of a given point the nearest one that has a small modular square

trace identities for commutators with applications to the distribution of eigenvalues we prove trace identities for commutators of operators which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic schrodinger operators and schrodinger operators on immersed manifolds in particular we prove bounds on the eigenvalue lambda n 1 in terms of the lower spectrum bounds on ratios of means of eigenvalues and universal monotonicity properties of eigenvalue moments which imply sharp versions of lieb thirring inequalities in the case of a schrodinger operator on an immersed manifold of dimension d we derive a version of reilly s inequality bounding the eigenvalue lambda n 1 of the laplace beltrami operator by a universal constant times vertical bar vertical bar h vertical bar n 2 infinity 2 d

reflection coefficient beyond all orders for singular problems ii closed spaced critical points on the nearest critical level line a crucial connection formula with closed spaced critical points on the nearest critical level line is established in this paper in order to use the generalized pokrovskii khalatnikov method to calculate the nontrivial behavior of the reflection coefficient for singular problems

on the third critical field in ginzburg landau theory using recent results by the authors on the spectral asymptotics of the neumann laplacian with magnetic field we give precise estimates on the critical field h c 3 describing the appearance of superconductivity in superconductors of type ii furthermore we prove that the local and global definitions of this field coincide near h c 3 only a small part near the boundary points where the curvature is maximal of the sample carries superconductivity we give precise estimates on the size of this zone and decay estimates in both the normal to the boundary and parallel variables

exponentially small adiabatic invariant for the schrodinger equation we study an adiabatic invariant for the time dependent schrfdinger equation which gives the transition probability across a gap from time t to time t when the hamiltonian depends analytically on time and t oo t oo we give sufficient conditions so that this adiabatic invariant tends to zero exponentially fast in the adiabatic limit

on time dependent schrodinger equations global well posedness and growth of sobolev norms in this paper we consider time dependent schrodinger linear pdes of the form i t l t where l t is a continuous family of self adjoint operators we give conditions for well posedness and polynomial growth for the evolution in abstract sobolev spaces if l t h v t where v t is a perturbation smooth in time and h is a self adjoint positive operator whose spectrum can be enclosed in spectral clusters whose distance is increasing we prove that the sobolev norms of the solution grow at most as t when t for any 0 if v t is analytic in time we improve the bound to log t for some 0 the proof follows the strategy due to howland joye and nenciu of the adiabatic approximation of the flow we recover most of known results and obtain new estimates for several models including 1 degree of freedom schrodinger operators on r and schrodinger operators on zoll manifolds

quantum phase corrections from adiabatic iteration the phase change gamma acquired by a quantum state psi t rangle driven by a hamiltonian h 0 t which is taken slowly and smoothly round a cycle is given by a sequence of approximants gamma k obtained by a sequence of unitary transformations the phase sequence is not a perturbation series in the adiabatic parameter epsilon because each gamma k except gamma 0 contains epsilon to infinite order for spin frac 1 2 systems the iteration can be described in terms of the geometry of parallel transport round loops c k on the hamiltonian sphere non adiabatic effects transitions must cause the sequence of gamma k to diverge for spin systems with analytic h 0 t this happens in a universal way the loops c k are sinusoidal spirals which shrink as epsilon k until k sim epsilon 1 and then grow as k the smallest loop has a size exp 1 epsilon comparable with the non adiabaticity

faraday effect revisited sum rules and convergence issues this is the third paper of a series revisiting the faraday effect the question of the absolute convergence of the sums over the band indices entering the verdet constant is considered in general sum rules and traces per unit volume play an important role in solid state physics and they give rise to certain convergence problems widely ignored by physicists we give a complete answer in the case of smooth potentials and formulate a number of open problems related to less regular perturbations

precise coupling terms in adiabatic quantum evolution the generic case for multi level time dependent quantum systems one can construct superadiabatic representations in which the coupling between separated levels is exponentially small in the adiabatic limit based on results from bete1 for special hamiltonians we explicitly determine the asymptotic behavior of the exponentially small coupling term for generic two state systems with real symmetric hamiltonian the superadiabatic coupling term takes a universal form and depends only on the location and the strength of the complex singularities of the adiabatic coupling function our proof is based on a new norm which allows to rigorously implement darboux principle a heuristic guideline widely used in asymptotic analysis

spectral analysis and stability properties of a relativistic deformation of the harmonic oscillator the spectral properties are established and the stability of the eigenvalues under the singular perturbation representing the relativistic corrections are proven for a family of ordinary differential operators describing a relativistic deformation of the quantum harmonic oscillator

the adiabatic limit of schrodinger operators on fibre bundles we consider schrodinger operators h delta g varepsilon v on a fibre bundle m mathop rightarrow limits pi b with compact fibres and a metric g varepsilon that blows up directions perpendicular to the fibres by a factor varepsilon 1 gg 1 we show that for an eigenvalue lambda of the fibre wise part of h satisfying a local gap condition and every n in mathbb n there exists a subspace of l 2 m that is invariant under h up to errors of order varepsilon n 1 the dynamical and spectral features of h on this subspace can be described by an effective operator on the fibre wise lambda eigenspace bundle mathcal e rightarrow b giving detailed asymptotics for h

lipshitz continuity of gap boundaries for hofstadter like spectra we consider an effective hamiltonianh representing the motion of a single band two dimensional electron in a uniform magnetic field thenh belongs to the rotation algebra namely the algebra of continuous functions over a non commutative 2 torus we define a non commutative analog of smooth functions by mean of elements of classc l n wherel andn characterize respectively the degree of differentiability with respect to the magnetic field and the torus variables we show that ifh is of classc 1 3 e the gap boundaries of the spectrum ofh are lipshitz continuous functions of the magnetic field at each point for which the gap is open

feedback stabilization of affine in the control stochastic differential systems by the control lyapunov function method the purpose of this paper is to study the asymptotic stability in probability of affine in the control stochastic differential systems sufficient conditions for the existence of control lyapunov functions leading to the existence of stabilizing feedback laws which are smooth except possibly at the equilibrium point of the system are provided

proof of the landau zener formula in an adiabatic limit with small eigenvalue gaps we consider a smooth operator valued functionh t that has two isolated non degenerate eigenvaluese a t ande b t for 0 we assume these eigenvalues are bounded away from the rest of the spectrum ofh t but have an avoided crossing with one another with a closest approach that iso as tends to zero under these circumstances we study the small e limit for the adiabatic schrodinger equation i varepsilon frac partial psi partial t h t varepsilon 1 2 psi we prove that the landau zener formula correctly describes the coupling between the adiabatic states associated with the eigenvaluese a t ande b t as the system propagates through the avoided crossing

a rigorous time bound for factoring integers in this paper a probabilistic algorithm is exhibited that factors any positive integer n into prime factors in expected time at most ln 2 1 o for n oo where l a b exp b logx a loglogx l a many practical factoring algorithms including the quadratic sieve and the elliptic curve method are conjectured to have an expected running time that satisfies the same bound but this is the first algorithm for which the bound can be rigorously proved nevertheless this does not close the gap between rigorously established time bounds and merely conjectural ones for factoring algorithms this is due to the advent of a new factoring algorithm the number field sieve which is conjectured to factor any positive integer n in time ln i 0 1 the algorithm analyzed in this paper is a variant of the class group relations method which makes use of class groups of binary quadratic forms of negative discriminant this algorithm was first suggested by seysen and later improved by a k lenstra who showed that the algorithm runs in expected time at most ln 12 1 o 1 if one assumes the generalized riemann hypothesis the main device for removing the use of the generalized riemann hypothesis from the proof is the use of multipliers in addition a character sum estimate for algebraic number fields is used with an explicit dependence on possible exceptional zeros of the corresponding l functions another factoring algorithm using class groups that has been proposed is the random class groups method it is shown that there is a fairly large set of numbers that this algorithm cannot be expected to factor as efficiently as had previously been thought department of mathematics university of california berkeley california 94720 e mail address hwl math berkeley edu department of mathematics university of georgia athens georgia 30602 e mail address carl ada math uga edu this content downloaded from 157 55 39 186 on tue 12 apr 2016 08 54 17 utc all use subject to http about jstor org terms

time development of exponentially small non adiabatic transitions optimal truncations of asymptotic expansions are known to yield approximations to adiabatic quantum evolutions that are accurate up to exponentially small errors in this paper we rigorously determine the leading order non adiabatic corrections to these approximations for a particular family of two level analytic hamiltonian functions our results capture the time development of the exponentially small transition that takes place between optimal states by means of a particular switching function our results confirm the physics predictions of sir michael berry in the sense that the switching function for this family of hamiltonians has the form that he argues is universal

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

non trivial prefactors in adiabatic transition probabilities induced by high order complex degeneracies the adiabatic limit of the transition probability for two level systems driven by real symmetric time dependent hamiltonians is considered when the hamiltonian depends analytically on time the transition probability is governed in simple cases by a complex eigenvalue crossing point which is generically a square root branching point for the eigenvalues in such a case the transition probability is given by the dykhne formula the recently discovered geometric prefactor being equal to one in this paper we deal with the general situation where the relevant eigenvalue crossing point is a branching point of order n 2 n or 1 for the eigenvalues and a zero of order m or 0 for the hamiltonian itself the analysis shows that the dykhne formula must be completed by a novel prefactor which depends on both n and m in particular this prefactor can take the value zero in contrast to the geometrical prefactor we also consider the case where the transition probability is governed by n complex eigenvalue crossing points of different orders nj and mj j 1 n the end result displays an interference phenomenon between the individual prefactors similar to the case of generic eigenvalue crossing points nj 1 and mj 0 considered earlier

factoring large numbers with a quadratic sieve the quadratic sieve algorithm was used to factor a 47 digit number into primes a comparison with wagstaff s results using the continued fraction early abort algorithm suggests that qs should be faster than cfea when the number being factored exceeds 60 digits plus or minus ten or more digits depending on details of the hardware and software

on the partial equiasymptotic stability in functional differential equations abstract a system of functional differential equations with delay dz dt z t zt where z is the vector valued functional is considered it is supposed that this system has a zero solution z 0 definitions of its partial stability partial asymptotical stability and partial equiasymptotical stability are given theorems on the partial equiasymptotical stability are formulated and proved

quantum degeneracy and classical perturbation theory the degenerate rayleigh schrodinger perturbation theory for the eigenvalues of the class of schrodinger operators s e s0 ev where s0 is the laplace beltrami operator on the one dimensional torus t1 and v is an analytic potential perturbed rotator is explicitly constructed by means of the algorithm of the classical resonant birkhoff normal form as a consequence the degenerate quantum perturbation theory tends to the classical one in the classical limit

cyclic thermodynamic processes and entropy production we study the time evolution of a periodically driven quantum mechanical system coupled to several reservoirs of free fermions at different temperatures this is a paradigm of a cyclic thermodynamic process we introduce the notion of a floquet liouvillean as the generator of the dynamics of the coupled system on an extended hilbert space we show that the time periodic state which the state of the coupled system converges to after very many periods corresponds to a zero energy resonance of the floquet liouvillean we then show that the entropy production per cycle is strictly positive a property that implies carnot s formulation of the second law of thermodynamics

histories of adiabatic quantum transitions the way in which the transition amplitude to an initially unoccupied state increases to its exponentially small final value is studied in detail in the adiabatic approximation for a 2 state quantum system by transforming to a series of superadiabatic bases clinging ever closer to the exact evolving state it is shown that transition histories renormalize onto a universal one in which the amplitude grows to its final value as an error function rather than via large oscillations as in the ordinary adiabatic basis the time for the universal transition is of order surd hslash delta where delta is the small adiabatic slowness parameter in perturbation theory the pre exponential factor of the final amplitude renormalizes superadiabatically from the incorrect value textstyle frac 1 3 pi for the ordinary adiabatic basis to the correct value unity the various histories could be observed in spin experiments

temporal exponential decay for the stark effect in atoms abstract let h 1 i 1 n i v x i 1 i x i x j 1 v x i n x y 1 y d y with a normalized gaussian suppose e 0 and that h h 1 e i 1 n x i has no eigenfunctions in l 2 r 3n if h 1 with ess h 1 then e ith decays exponentially at a rate governed by the positions of the resonances of h

semiclassical limit for multistate klein gordon systems almost invariant subspaces and scattering theory by using the method of helffer and sjostrand to construct moyal projections we extend the almost invariant subspace theory to the semiclassical context applications to the semiclassical limit for two component klein gordon hamiltonian are given more precisely under the conditions that the potential is analytic and its eigenvalues never cross we prove that the scattering matrix is block diagonal up to exponentially small errors also we show how the existence of almost invariant subspaces leads to the existence of quasimodes with exponentially long lifetimes

low lying spectral gaps induced by slowly varying magnetic fields abstract we consider a periodic schrodinger operator in two dimensions perturbed by a weak magnetic field whose intensity slowly varies around a positive mean we show in great generality that the bottom of the spectrum of the corresponding magnetic schrodinger operator develops spectral islands separated by gaps reminding of a landau level structure first we construct an effective hofstadter like magnetic matrix which accurately describes the low lying spectrum of the full operator the construction of this effective magnetic matrix does not require a gap in the spectrum of the non magnetic operator only that the first and the second bloch eigenvalues do not cross but their ranges might overlap the crossing case is more difficult and will be considered elsewhere second we perform a detailed spectral analysis of the effective matrix using a gauge covariant magnetic pseudo differential calculus adapted to slowly varying magnetic fields as an application we prove in the overlapping case the appearance of spectral islands separated by gaps

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

on asymptotic perturbation theory for quantum mechanics almost invariant subspaces and gauge invariant magnetic perturbation theory singular perturbation theory for quantum mechanics is considered in a framework generalizing the spectral concentration theory under very general conditions asymptotic estimations on the rayleigh schrodinger expansions of the perturbed spectral projections are obtained as a consequence almost invariant subspaces of exponential order are constructed the results cover practically all singular perturbations considered in nonrelativistic quantum mechanics in the magnetic field case under the condition that the magnetic field does not increase at infinity a gauge invariant perturbation theory leading to convergent series with field dependent coefficients is developed

a converse lyapunov theorem for almost sure stabilizability we prove a converse lyapunov theorem for almost sure stabilizability and almost sure asymptotic stabilizability of controlled diffusions given a stochastic system a s stochastic open loop stabilizable at the origin we construct a lower semicontinuous positive definite function whose level sets form a local basis of viable neighborhoods of the equilibrium this result provides with the direct lyapunov theorems proved in a companion paper a complete lyapunov like characterization of the a s stabilizability

double wells nevanlinna analyticity distributional borel sum and asymptotics we consider the energy levels of a stark family in the parameterj of quartic double wells with perturbation parameterg h g j p 2 x 2 1 gx 2 j left gx frac 1 2 right for non evenj and for the symmetric casej 0 we prove analyticity in the full nevanlinna disk g 2 r 1 of theg2 plane as predicted by crutchfield by means of an approximation we give a heuristic estimate of the asymptotic smallg behaviour showing the relation between the avoided crossings and the failure of the usual perturbation series

trace asymptotics formula for the schrodinger operators with constant magnetic fields abstract in this paper we consider the 2d schrodinger operator with constant magnetic field h v d x b y 2 d y 2 v h x y where v tends to zero at infinity and h is a small positive parameter we will be concerned with two cases the semi classical limit regime v h x y v h x h y and the large coupling constant limit case v h x y h v x y we obtain a complete asymptotic expansion in powers of h 2 of tr h v h where h c 0 r r we also give a weyl type asymptotics formula with optimal remainder estimate of the counting function of eigenvalues of h v

spectral stability under tunneling we study the spectral properties of multiple well schrodinger operators on n we give in particular upper bounds on energy shifts due to tunnel effect and localization properties of wave packets our methods are based on agmon type estimates for resolvents in classically forbidden regions and geometric perturbation theory our results are valid also for an infinite number of wells arbitrary spectral type and in non semi classical regimes

precise exponential estimates in adiabatic theory general adiabatic evolutions associated to hamiltonians which admit a holomorphic extension with respect to the time variable in a complex strip and whose spectrum satisfies a gap condition are studied an explicit rate of exponential decay is given which is related to simple geometric quantities associated to the spectrum of the hamiltonian for the transition probability between the two parts of the spectrum when the evolution is taken from to

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

canonical expansion of symmetric operators and perturbation theory let h be any symmetric schrodinger operator of the type 2 x21 x2d igw x1 xd on where w is any odd homogeneous polynomial and it is proved that is self adjoint and that its eigenvalues coincide up to a sign with the singular values of h i e the eigenvalues of moreover we explicitly construct the canonical expansion of h and determine the singular values j of h through the borel summability of their divergent perturbation theory the singular values yield estimates of the location of the eigenvalues j of h by weyl s inequalities

spectral and propagation results for magnetic schrodinger operators a c algebraic framework abstract we study generalized magnetic schrodinger operators of the form h h a v h a v where h is an elliptic symbol a i a with a a vector potential defining a variable magnetic field b and v is a scalar potential we are mainly interested in anisotropic functions b and v the first step is to show that these operators are affiliated to suitable c algebras of magnetic pseudodifferential operators a study of the quotient of these c algebras by the ideal of compact operators leads to formulae for the essential spectrum of h h a v expressed as a union of spectra of some asymptotic operators supported by the quasi orbits of a suitable dynamical system the quotient of the same c algebras by other ideals give localization results on the functional calculus of the operators h h a v which can be interpreted as non propagation properties of their unitary groups

on sharp transitions in making squares in the fastest performing integer factoring algorithms one creates a sequence of integers in a pseudo random way and wishes to rapidly determine a subsequence whose product is a square in 1994 pomerance stated the following problem which encapsulates all of the key issues select integers a1 a2 at random from the interval 1 x until some nonempty subsequence has product equal to a square find a good estimate for the expected stopping time of this process a good solution should allow one to determine the optimal choice of parameters in many factoring algorithms pomerance 1994 using an idea of schroeppel 1985 showed that with probability 1 o 1 the rst subsequence whose product equals a square

power law corrections to the kubo formula vanish in quantum hall systems in first order perturbation theory conductivity is given by the kubo formula which in a quantum hall system equals the first chern class of a vector bundle we apply the adiabatic theorem to show that these topological constraints quantize the averaged conductivity to all orders of perturbation theory hence the kubo formula is valid to all orders

distributional borel summability of odd anharmonic oscillators it is proved that the divergent rayleigh schrodinger perturbation expansions for the eigenvalues of any odd anharmonic oscillator are borel summable in the distributional sense to the resonances naturally associated with the system

geometric theory of stark resonances in multielectron systems in this paper we consider a class of many body systems in a weak homogeneous electric field this class includes atoms and molecules with infinitely heavy nuclei it follows from one of the results of this paper and a result of s 3 that the bound states of such systems in the absence of electric field turn into resonances which we call the stark resonances as soon as the electric field is switched on the stability part of this result was earlier proven in hesi see also hu 2 under an assumption of dilation analyticity the main result of this paper is exponential bounds on the width and therefore the lower exponential bounds on the life time of the stark resonances these bounds are given in terms of the stark instanton action in contrast to the usual one body action the latter is not entirely classical but incorporates certain quantum data like ionization energies the bounds give a partial generalization to the many electron case of the well known oppenheimer formula for the hydrogen

geometric currents in piezoelectricity as a simple model for piezoelectricity we consider a gas of infinitely many non interacting electrons subject to a slowly time dependent periodic potential we show that in the adiabatic limit the macroscopic current is determined by the geometry of the bloch bundle as a consequence we obtain the king smith and vanderbilt formula up to errors smaller than any power of the adiabatic parameter

a semiclassical egorov theorem and quantum ergodicity for matrix valued operators we study the semiclassical time evolution of observables given by matrix valued pseudodifferential operators and construct a decomposition of the hilbert space l2 open image in new windowd open image in new windown into a finite number of almost invariant subspaces for a certain class of observables that is preserved by the time evolution we prove an egorov theorem we then associate with each almost invariant subspace of l2 open image in new windowd open image in new windown a classical system on a product phase space t open image in new windowd open image in new window where open image in new window is a compact symplectic manifold on which the classical counterpart of the matrix degrees of freedom is represented for the projections of eigenvectors of the quantum hamiltonian to the almost invariant subspaces we finally prove quantum ergodicity to hold if the associated classical systems are ergodic

a probabilistic factorization algorithm with quadratic forms of negative discriminant we propose a probabilistic algorithm for factorization of an integer n with run time expvlog n log log n v54 1 asymptotically our algorithm will be as fast as the wellknown factorization algorithm of morrison and brillhart the latter algorithm will fail in several cases and heuristic assumptions are needed for its run time analysis our new algorithm will be analyzed under the assumption of the extended riemann hypothesis and it will be of las vegas type on input n the new algorithm will factor n with probability 12 in case of prime n the algorithm will prove the primality of n with probability 2 introduction until the last decade the centuries old problem of factoring integers was mainly a problem for specialists worldwide interest in factoring integers increased dramatically in 1978 when rivest shamir and adleman 32 published their public key cryptosystem whose security relies on the fact that some large integers are hard to factor gauss 9 already discovered a close connection between the factorization of a natural number n and the theory of quadratic forms of discriminant 4n now quadratic forms are one of the most important tools for factoring integers examples for efficient factorization algorithms are among others the algorithms of morrison and brillhart 26 and lenstra and schnorr 22 the former works with the continued fraction expansion of fn which is closely related to the theory of quadratic forms of discriminant 4n see 20 while the latter works with quadratic forms of discriminant 4n at present the most efficient factorization algorithm is the quadratic sieve algorithm see 29 which can also be expressed in terms of quadratic forms for an overview of modem factorization algorithms we refer to the papers of guy 10 monier 25 and pomerance 29 a deeper understanding of the theory of quadratic forms is of great importance for the analysis of modem factorization algorithms in this paper we shall only deal with the theory of quadratic forms of negative discriminant which is considerably more simple than the theory of forms of positive discriminants using this theory we obtain a probabilistic factorization algorithm with run time exp vlog n log log n v4 in this paper we denote by log x the natural logarithm of x we have 5 4 1 118 received may 17 1985 revised june 12 1986 1980 mathematics subject classification primary 10a30 68c25 1987 american mathematical society 0025 5718 87 1 00 25 per page

gaussian decay for a difference of traces of the schrodinger semigroup associated with the isotropic harmonic oscillator abstract this paper deals with the derivation of a sharp estimate on the difference of traces of the one parameter schrodinger semigroup associated with the quantum isotropic harmonic oscillator denoting by h the self adjoint realization in l 2 r d d 1 2 3 of the schrodinger operator 1 2 1 2 2 x 2 0 and by h l l 0 the dirichlet realization in l 2 l d where l d x r d l 2 x l l 2 l 1 d we prove that the difference of traces tr l 2 r d e t h tr l 2 l d e t h l t 0 has for l sufficiently large a gaussian decay in l furthermore the estimate that we derive is sharp in the two following senses its behavior when t 0 is similar to the one given by tr l 2 r d e t h 2 sinh 2 t d and the exponential decay in t arising from tr l 2 r d e t h when t is preserved for illustrative purposes we give a simple application within the framework of quantum statistical mechanics

universal lower bounds on eigenvalue splittings for one dimensional schrodinger operators we provide lower bounds on the eigenvalue splitting for d2 dx2 v x depending only on qualitative properties ofv for example ifv is c on a b ande n en 1 are two successive eigenvalues of d2 dx2 v withu a u b 0 boundary conditions and if lambda mathop max limits e in e n 1 e n x in a b e v x 1 2 then e n e n 1 geqq pi lambda 2 exp left lambda b a right the exponential factor in such bounds are saturated precisely in tunneling examples our results arenot restricted tov s of compact support but only require e n mathop lim limits overline x to infty v x

the faraday effect revisited thermodynamic limit abstract this paper is the second in a series revisiting the effect of faraday rotation we formulate and prove the thermodynamic limit for the transverse electric conductivity of bloch electrons as well as for the verdet constant the main mathematical tool is a regularized magnetic and geometric perturbation theory combined with elliptic regularity and agmon combes thomas uniform exponential decay estimates

full asymptotic expansion of transition probabilities in the adiabatic limit the authors consider a two level quantum mechanical system driven by an analytic time dependent hamiltonian of the form h in t in the adiabatic limit in or 0

lyapunov s second method for random dynamical systems abstract the method of lyapunov functions lyapunov s second or direct method was originally developed for studying the stability of a fixed point of an autonomous or non autonomous differential equation it was then extended from fixed points to sets from differential equations to dynamical systems and to stochastic differential equations we go one step further and develop lyapunov s second method for random dynamical systems and random sets together with matching notions of attraction and stability as a consequence lyapunov functions will also be random our test is that the extension be coherent in the sense that it reduces to the deterministic theory in case the noise is absent and that we can prove that a random set is asymptotically stable if and only if it has a lyapunov function several examples are treated including the stochastic lorenz system

instability of nonlinear dispersive solitary waves we consider linear instability of solitary waves of several classes of dispersive long wave models they include generalizations of kdv bbm regularized boussinesq equations with general dispersive operators and nonlinear terms we obtain criteria for the existence of exponentially growing solutions to the linearized problem the novelty is that we dealt with models with nonlocal dispersive terms for which the spectra problem is out of reach by the evans function technique for the proof we reduce the linearized problem to study a family of nonlocal operators which are closely related to properties of solitary waves a continuation argument with a moving kernel formula is used to find the instability criteria these techniques have also been extended to study instability of periodic waves and of the full water wave problem

monte carlo factoring algorithm with finite storage we present an algorithm which will factor an integer n quite efficiently if the class number h n is free of large prime divisors the running time t n number of compositions in the class group satisfies prob t m n1 2r r 2 r 2 for random me n 2 n and r 2 so far it is unpredictable which numbers will be factored fast running the algorithm on all integers n s with s rr and r sqrt ln n mathord left vphantom ln n ln ln n right kern nulldelimiterspace ln ln n every composite integer n will be factored in o left exp sqrt ln n mathord left vphantom ln n ln ln n right kern nulldelimiterspace ln ln n right bit operations and fixed amount of storage space in our analysis we assume a lower bound on the frequency of class numbers h m m n which are free of large prime divisors

reflection coefficient beyond all orders for singular problems this paper generalizes the pokrovskii khalatnikov method to calculate the actual behavior of the reflection coefficient for singular problems which vanishes to all orders in the small parameter e two different classes of reflection coefficient problems are considered

effective dynamics for bloch electrons peierls substitution and beyond we consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential x and vector potential a x with x d and 1 we prove that associated to an isolated family of bloch bands there exists an almost invariant subspace of l 2 d and an effective hamiltonian governing the evolution inside this subspace to all orders in to leading order the effective hamiltonian is given through the peierls substitution we explicitly compute the first order correction from a semiclassical analysis of this effective quantum hamiltonian we establish the first order correction to the standard semiclassical model of solid state physics

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

an analytic family of cal p cal t symmetric hamiltonians with real eigenvalues we study nonlinear perturbative expansions for symmetric local schrodinger operators the schrodinger operator is a sum of the harmonic oscillator hamiltonian and a local symmetric potential depending in general nonlinearly on the perturbation parameter a specific class of models having real spectrum for any value of the parameter is proposed

stability of schrodinger eigenvalue problems we derive a general stability criterion for discrete eigenvalues of schrodinger operators such asa p 2 v x using only strong continuity ofa anda in the perturbation parameter the theory is developed for non selfadjoint operators and illustrated with examples like the anharmonic oscillator the stark and the zeeman effect the principal tools are weyl s criterion for the essential spectrum and a construction due to enss 5 they are also used to extend the classical invariance theorems for the essential spectrum to certain singular perturbations including some local perturbations of the laplacian by differential operators of arbitrary high order

perturbation theory of symmetric hamiltonians in the framework of perturbation theory the reality of the perturbed eigenvalues of a class of symmetric hamiltonians is proved using stability techniques we apply this method to symmetric unperturbed hamiltonians perturbed by symmetric additional interactions

an improved stability result for resonances we prove stability of shape resonances for the sequence of schrodinger equations d2 dx2 u x wn x 4 x e4 x 0 s x oo in the limit n oo where the barrier potentials wn x are integrable nonnegative supported in the interval 1 a 1 a oo and approach infinity pointwise a e for x e 1 a as n oo in the course of our investigation we prove that for suitable complex initial conditions the solution to the riccati equation s x wn x e s x 2 goes to 0 as n oo uniformly on compact subsets of 1 a our approach is via ordinary differential equations using outgoing wave boundary conditions to define resonances our stability result extends a similar result of ashbaugh and harrell who use an argument based on asymptotics and the implicit function theorem to study the above problem with xv x replacing wn x our approach is to use the riccati equation analysis mentioned above and an application of hurwitz s theorem from complex variable theory 1 introduction the behavior of eigenvalues of schrodinger operators has been the subject of extensive investigation since the advent of quantum mechanics in particular perturbation theory has been on a firm mathematical footing since the work of rellich in the thirties 5 kato 6 and sz nagy 7 also made important contributions in contrast the study of resonances has received comparatively little attention only rather recently have some of the most basic properties of eigenvalues been extended to resonances 8 10 this disparity is not surprising in view of the relative difficulty of studying resonances in a hilbert space framework whereas the eigenvalues of a schrodinger operator can be dealt with directly in terms of the given selfadjoint operator at best resonances can be viewed as eigenvalues of certain associated nonselfadjoint operators a result established by simon 8 for the case of dilation analytic potentials in the one dimensional or separable multi dimensional