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

gamow vectors and borel summability in a class of quantum systems we analyze the detailed time dependence of the wave function x t for one dimensional hamiltonians h partial x 2 v x where v for example modeling barriers or wells and x 0 are compactly supported

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

adiabatic passage for a lossy two level quantum system by a complex time method using a complex time method with the formalism of stokes lines we establish a generalization of the davis dykhne pechukas formula which gives in the adiabatic limit the transition probability of a lossy two state system driven by an external frequency chirped pulse shaped field the conditions that allow this generalization are derived we illustrate the result with the dissipative allen eberly and rosen zener models

smoothing of stokes discontinuities when the asymptotics of an analytic function f z are given in terms of two asymptotic forms s 1 z s 2 z alternately dominant in alternating sectors of the z plane stokes observed that the coefficients of s 1 s 2 while constant in any one sector can change from sector to sector and that to a first approximation the change appears to be discontinuous recently berry has given an interesting formal argument which shows in more detail how the change takes place and the present paper gives a rigorous treatment of berry s result

linear time dependent hamiltonian systems beyond the adiabatic limit a classical version of the magnus expansion well suited to studying adiabatic time evolution is built up the method improves the adiabatic approximation while being symplectic in character it is shown that the first order approximation is already accurate enough even far from the adiabatic limit an analysis of the changes suffered by the adiabatic invariant of a linear hamiltonian system along its time evolution illustrates part of the above results asymptotic formulae for such changes are also obtained with explicit computation of pre exponential factors

nonadiabatic transitions for a decaying two level system geometrical and dynamical contributions we study the landau zener problem for a decaying two level system described by a non hermitian hamiltonian depending analytically on time use of a super adiabatic basis allows us to calculate the survival probability p in the slow sweep limit without specifying the hamiltonian explicitly it is found that p consists of a dynamical and a geometrical factor the former is determined by the complex adiabatic eigenvalues e t only whereas the latter solely requires the knowledge of t the ratio of the components of each of the adiabatic eigenstates both factors can be split into a universal one depending only on the complex level crossing points and a nonuniversal one involving the full time dependence of e t and t this general result is applied to the akulin schleich model where the initial upper level is damped with damping constant for analytic power law sweeps we find that stuckelberg oscillations of p exist for smaller than a critical value c and disappear for c a physical interpretation of this behaviour will be presented by use of a damped harmonic oscillator

overlapping stokes smoothings survival of the error function and canonical catastrophe integrals we derive doubly uniform approximations for the remainder in the optimally truncated saddle point expansion for an integral containing a large parameter double uniformity means that the formulae remain valid while distant saddles responsible for the divergence of the expansion coalesce and separate as described by catastrophe theory and while the subdominant exponentials they contribute switch on and off as described by the error function smoothing of the stokes phenomenon two sorts of asymptotic singularity are thereby united in a common framework the formula for the remainder incorporates both the stokes error function and the canonical catastrophe integrals a numerical illustration is given in which the distant cluster contains two saddles the asymptotic theory gives an accurate description of the details of the fractional remainder even when this is of order exp 36

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

stieltjes transforms and the stokes phenomenon recently berry olver and jones have found uniform asymptotic expansions for the exponentially small remainder terms that result when asymptotic expansions are optimally truncated these uniform expansions describe the rapid change in the behaviour of the remainders as a stokes line is crossed we show how such uniform expansions may be found when a function can be expressed as a stieltjes transform such an approach has the following advantages the uniform expansion is calculated directly non uniform expansions away from the stokes line are readily found and explicit error bounds may be established we illustrate the method by application to the modified bessel function k v z

principal vertex operator representations for toroidal lie algebras we introduce the principal vertex operator representations for the toroidal lie algebras generalizing the construction for the affine kac moody algebras we also represent the derivations of the toroidal algebras and introduce analogs of the sugawara operators

unitary transformations depending on a small parameter we formulate a unitary perturbation theory for quantum mechanics inspired by the liedeprit formulation of canonical transformations the original hamiltonian is converted into a solvable one by a transformation obtained through a magnus expansion this ensures unitarity at every order in a small parameter a comparison with the standard perturbation theory is provided we work out the scheme up to order ten with some simple examples

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

convergence from divergence we show how to convert divergent series which typically occur in many applications in physics into rapidly convergent inverse factorial series this can be interpreted physically as a novel resummation of perturbative series being convergent these new series allow rigorous extrapolation from an asymptotic region with a large parameter to the opposite region where the parameter is small we illustrate the method with various physical examples and discuss how these convergent series relate to standard methods such as borel summation and also how they incorporate the physical stokes phenomenon we comment on the relation of these results to dyson s physical argument for the divergence of perturbation theory this approach also leads naturally to a wide class of relations between bosonic and fermionic partition functions and klein gordon and dirac determinants

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 optimal truncation of divergent series solutions of nonlinear differential systems we prove that for divergent series solutions of nonlinear or linear differential systems near a generic irregular singularity the common prescription of summation to the least term is if properly interpreted meaningful and correct and we extend this method to trans series solutions in every direction in the complex plane at the singularity stokes9 directions not excepted there exists a non empty set of solutions whose difference from the optimally i e near the least term truncated asymptotic series is of the same exponentially small order of magnitude as the least term of the series there is a family of generalized borel summation formulae b which commute with the usual algebraic and analytic operations addition multiplication differentiation etc we show that there is exactly one of them b 0 such that for any formal series solution f b 0 f differs from the optimal truncation of f by at most the order of the least term of f we show in addition that the berry smoothing phenomenon is universal within this class of differential systems whenever the terms beyond all orders change in crossing a stokes line these terms vary smoothly on the berry scale x x 1 2 and the transition is always given by the error function under the same conditions we show that dingle9 rule of signs for stokes9 transitions holds

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

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

the bloch wave operator generalizations and applications part i the time independent case this is part 1 of a two part review on wave operator theory and methods the basic theory of the time independent wave operator is presented in terms of partitioned matrix theory for the benefit of general readers with a discussion of the links between the matrix and projection operator approaches the matrix approach is shown to lead to simple derivations of the wave operators and effective hamiltonians of lowdin bloch des cloizeaux and kato as well as to some associated variational forms the principal approach used throughout stresses the solution of the nonlinear equation for the reduced wave operator leading to the construction of the effective hamiltonians of bloch and of des cloizeaux several mathematical techniques which are useful in implementing this approach are explained some of them being relatively little known in the area of wave operator calculations the theoretical discussion is accompanied by several specimen numerical calculations which apply the described techniques to a selection of test matrices taken from the previous literature on wave operator methods the main emphasis throughout is on the use of numerical methods which use iterative or perturbation algorithms with simple pade approximant methods being found sufficient to deal with most of the cases of divergence which are encountered the use of damping factors and relaxation parameters is found to be effective in stabilizing calculations which use the energy dependent effective hamiltonian of lowdin in general the computations suggest that the numerical applications of the nonlinear equation for the reduced wave operator are best carried out with the equation split into a pair of equations in which the bloch effective hamiltonian appears as a separate entity the presentation of the theoretical and computational details throughout is accompanied by references to and discussion of many works which have used wave operator methods in physics chemistry and engineering some of the techniques described in this part 1 will be further extended and applied in part 2 of the review which deals with the changes which are required to extend wave operator theory to the case of a time dependent hamiltonian such as that which describes the interaction of a laser pulse with an atom or molecule

superadiabatic tracking of quantum evolution the adiabatic quantum evolution of a two state system without energy level crossings is an example of the stokes phenomenon in the latter a small subdominant exponential is an asymptotic expansion appears when a stokes line is crossed truncating the dominant asymptotic series at its least term causes the multiplier of the subdominant term to rise in a smooth compact and universal manner across the stokes line in quantum evolution this corresponds to a smooth transition universal in form between superadiabatic basis states high order wkb approximate solutions of the time dependent schrodinger equation the authors give a numerical demonstration of this previously predicted universality by constructing for two hamiltonians the superadiabatic quantum bases asymptotic to the actual evolving state universality when a stokes line is crossed is seen in the changing probability that the system makes a transition away from the superadiabatic state and occurs at that order of superadiabatic approximation corresponding to truncating the asymptotic series at its least term

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

twisted logarithmic modules of free field algebras given a non semisimple automorphism of a vertex algebra v the fields in a twisted v module involve the logarithm of the formal variable and the action of the virasoro operator l0 on such a module is not semisimple we construct examples of such modules and realize them explicitly as fock spaces when v is generated by free fields specifically we consider the cases of symplectic fermions odd superbosons free fermions and system even superfermions in each case we determine the action of the virasoro algebra

smoothing of the stokes phenomenon for high order differential equations we extend the class of functions for which the smooth transition of a stokes multiplier across a stokes line can be rigorously established to functions satisfying a certain differential equation of arbitrary order n the equation chosen admits solutions of hypergeometric function type which in the case n 2 are related to the parabolic cylinder functions in general the solutions of this equation involve compound asymptotic expansions valid in certain sectors of the complex z plane with more than one dominant and subdominant series the functional form of the stokes multipliers expressed in terms of an appropriately scaled variable describing transition across a stokes line is found to obey the error function smoothing law derived by berry

distribution of work in isothermal nonequilibrium processes diffusive motion in an externally driven potential is considered it is shown that the distribution of work required to drive the system from an initial equilibrium state to another is gaussian for slow but finite driving our result is obtained by projection method techniques exploiting a small parameter defined as the switching rate between the two states of the system the exact solution for a simple model system shows that such an expansion may fail in higher orders since the mean and the variance following from the exact distribution show nonanalytic behavior

irreducible representations for toroidal lie algebras abstract in this work a large number of irreducible representations with finite dimensional weight spaces are constructed for some toroidal lie algebras to accomplish this we develop a general theory of z n graded lie algebras with polynomial multiplication we construct modules by the standard inducing procedure and study their irreducible quotients using the vertex operator technics

optical uniform estimates and rigorous asymptotics beyond all orders for a class of ordinary differential equations for first order differential equations of the form y9 p p x y p and second order homogeneous linear differential equations y a x y9 b x y 0 with locally integrable coefficients having asymptotic possibly divergent power series when x oo on a ray arg x const under some further assumptions it is shown that on the given ray there is a one to one correspondence between true solutions and complete formal solutions the correspondence is based on asymptotic inequalities which are required to be uniform in x and optimal with respect to certain weights

on a preemptive markovian queue with multiple servers and two priority classes we consider a queueing system with multiple servers and two classes of customers operating under a preemptive resume priority rule the arrival process for each class is poisson and the service times are exponentially distributed with different means in a convenient state space representation of the system we obtain the matrix equation for the two dimensional vector valued generating function of the equilibrium probability distribution we give a rigorous proof that by successively eliminating variables from the matrix equations a nonsingular block tridiagonal system of equations is obtained for the set of m 1m 2 2 constants that describe the probabilities of the states of the system when no customers are awaiting service the mean waiting time of the low priority customers is shown to be given by a simple formula in terms of the known waiting time of the high priority customers and the expected number of low priority customers in the queue when no high priority customers are waiting

slowly changing potential problems in quantum mechanics adiabatic theorems ergodic theorems and scattering we employ the recently developed multi time scale averaging method to study the large time behavior of slowly changing in time hamiltonians we treat some known cases in a new way such as the zener problem and we give another proof of the adiabatic theorem in the gapless case we prove a new uniform ergodic theorem for slowly changing unitary operators this theorem is then used to derive the adiabatic theorem do the scattering theory for such hamiltonians and prove some classical propagation estimates and asymptotic completeness

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

exact analytical solution for coupled time independent schrodinger equations with certain model potentials an exact analytical solution is obtained for coupled time independent schrodinger equations with model potentials and this is made possible by solving a single fourth order ordinary differential equation derived from the original coupled equations exact closed form solutions for the non adiabatic transition matrices or scattering matrices are found for scattering boundary conditions with three two and one open channels respectively how to apply the present results to deal with general potentials is also briefly analysed

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

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

vertex representations for the ν 1 toroidal lie algebra of type bl abstract we give vertex operator constructions for the toroidal lie algebra of type b l 1 l 2 we prove that the subquotients of the modules are completely reducible and give the irreducible decomposition

superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum during the last years several theoretical and experimental works have been devoted to different aspects of the adiabatic regime of quantum dynamics we consider here two aspects of this physically important regime the construction of approximate evolutions of the schriidinger equation called superadiabatic evolutions and the explicit computation of the probability of a transition between two nondegenerate energy levels isolated in the spectrum of the hamiltonian from the mathematical point of view the adiabatic limit for time dependent systems is of the same kind as the semiclassical limit for stationary systems it is a singular limit which cannot be tackled by usual perturbation methods recently a mathematical theory of the two considered aspects has been developed when the time dependent hamiltonian is analytic in time the transition probability for two level systems has been studied in refs l 3 and was shown to be exponentially small in the adiabaticity parameter for general systems in ref 4 on the other hand the existence of superadiabatic evolutions has been shown in ref 5 however no general formula for the transition probability between isolated levels in the adiabatic limit can be found in the literature it is the goal of this paper to establish such a formula for general unbounded hamiltonians the paper is divided into two parts we first construct a superadiabatic evolution which approximates the true evolution of the schrijdinger equation up to exponentially small correction terms with respect to the parameter controlling the adiabatic regime once the superadiabatic evolution is defined we show that the computation of the transition probability of the initial problem reduces to an effective two dimensional problem which we analyze as in ref 1 the concept of superadiabatic evolution can be traced back up to a paper of garrido 6 who considered evolutions different from the usual adiabatic evolution which he used to obtain higher order corrections to the adiabatic theorem of quantum mechanics but it is berry7v8 who emphasized the importance of such evolutions in the present problem and introduced the

exact analysis of adiabatic invariants in the time dependent harmonic oscillator the theory of adiabatic invariants has a long history and important applications in physics here we treat exactly the general time dependent 1d harmonic oscillator which cannot be solved in general we follow the time evolution of an initial ensemble of phase points with sharply defined energy e0 and calculate rigorously the distribution of energy e1 after time t and all its moments especially its average value and variance 2 using wkb theory we get the exact result for the leading asymptotic behaviour of 2

the fourier transform of the quartic gaussian exp ax4 hypergeometric functions power series steepest descent asymptotics and hyperasymptotics and extensions to exp ax2n the fourier transform of a quartic gaussian f k exp ikx exp x 4 dx is important in the theory of radial basis functions with f r exp r 4 we show that the transform is the exact sum of two generalized hypergeometric functions 0f 2 this implies that f k is an entire function and also gives an analytical form for the power series coefficients of arbitrary degree using the method of steepest descent for integrals we derive the first five terms in an asymptotic series in powers of k 4 3 through a simple hyperasymptotic analysis we show that the magnitude of the error of the optimally truncated series superasymptotic error is oexp 0 818k 4 3 the lowest order approximation f k 2 7 6 p 3k 1 3exp 2 1 33 16 k 4 3 cos 3 3 22 1 3 16 k 4 3 p 6 shows that the transform f k is an exponentially decaying oscillation we show that the steepest descent method 2 27 bender and orszag 1978 miller 2006 26 33 lauwerier 1966 ursell 1965 can be applied to the transform of exp ax 2 n for general integer n and give the first two terms another theme is that modern computer technology has simplified the steepest descent method the steepest descent paths can be plotted by merely displaying the contours of the imaginary part of the phase function j t where z j t is the logarithm of the integrand and z is the large parameter here proportional to k 4 3 the task of inverting w t to obtain a series in powers of w for the metric factor dt dw after the usual change of integration variable can now be done to high order by a few lines of an algebraic manipulation language as illustrated by a short table with the complete maple code

rigorous bounds of stokes constants for some nonlinear ordinary differential equations at rank one irregular singularities a rigorous way of obtaining sharp bounds for stokes constants is introduced and illustrated on a concrete problem arising in applications

semiclassically weak reflections above analytic and non analytic potential barriers the coefficient r for reflection above a barrier v x is computed semiclassically i e as h cross to 0 employing an exact multiple reflection series whose mth term is a 2m 1 fold integral if v x is analytic all terms have the same semiclassical order exp h cross 1 the multiple integrals are evaluated exactly and the series summed if v x has a discontinuous nth derivative the term m 1 dominates semiclassically and gives r approximately h cross n if v x has all derivatives continuous but possesses an essential singularity on the real axis the term m 1 again dominates semiclassically and for v approximately exp mod x mod n gives r approximately exp h cross n n 1 with an oscillatory factor corresponding to transmission resonances the formulae are illustrated by computations of mod r mod 2 for four potentials with different continuity properties and show the limiting asymptotics emerging only when the de broglie wavelength is less than 1 of the barrier width and mod r mod 2 approximately 10 1000

nonadiabatic transitions through tilted avoided crossings we investigate the transition of a quantum wave packet through a one dimensional avoided crossing of molecular energy levels when the energy levels at the crossing point are tilted using superadiabatic representations and an approximation of the dynamics near the crossing region we obtain an explicit formula for the transition wave function our results agree extremely well with high precision ab initio calculations

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

superadiabatic transitions in quantum molecular dynamics we study the dynamics of a molecule s nuclear wave function near an avoided crossing of two electronic energy levels for one nuclear degree of freedom we derive the general form of the schrodinger equation in the nth superadiabatic representation for all n n using these results we obtain closed formulas for the time development of the component of the wave function in an initially unoccupied energy subspace when a wave packet travels through the transition region in the optimal superadiabatic representation which we define this component builds up monotonically finally we give an explicit formula for the transition wave function away from the avoided crossing which is in excellent agreement with high precision numerical calculations

stokes s phenomenon for superfactorial asymptotic series superfactorial series depending on a parameter z are those whose terms a n z grow faster than any power of n if the terms get smaller before they increase the function f z represented by sum 0 infty a n z will exhibit a stokes phenomenon similar to that occurring in asymptotic series whose divergence is merely factorial across stokes lines in the z plane where the late terms all have the same phase a small exponential switches on in the remainder when the series is truncated near its least term the jump is smooth and described by an error function whose argument has a slightly more general form than in the factorial case this result is obtained by a method which is heuristic but applies to superfactorial series where borel summation fails several examples are given including an analytical interpretation of the sum and a numerical test of the error function formula for the function represented by f z sum 0 infty exp n 2 a 2nz where a gg 1

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

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

tests for the nonergodicity of multidimensional markov chains a markov chain is ergodic if it has a proper steady state distribution the stability of a queueing system often reduces to the ergodicity of an underlying multidimensional chain we gave criteria for the nonergodicity of a multidimensional markov chain and a generalization to continuous time markov chains we also give queueing model examples

exponentially improved asymptotic solutions of ordinary differential equations ii irregular singularities of rank one re expansions are found for the optimal remainder terms in the well known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one the re expansions are in terms of generalized exponential integrals and have greater regions of validity than the original expansions as well being considerably more accurate and providing a smooth interpretation of the stokes phenomenon they are also of strikingly simple form in addition explicit asymptotic expansions for the higher coefficients of the original asymptotic solutions are obtained

normal forms of holomorphic matrix valued functions and corresponding forms for singular differential operators abstract normal forms of a holomorphic matrix valued function a z under different classes of similarity transformations and corresponding forms of the singular differential operator z r d dz a z r n r 1 are investigated the problem of holomorphic similarity of matrix valued functions is given special consideration

exponentially accurate semiclassical tunneling wavefunctions in one dimension we study the time behavior of wavefunctions involved in tunneling through a smooth potential barrier in one dimension in the semiclassical limit we determine the leading order component of the wavefunction that tunnels it is exponentially small in 1 for a wide variety of incoming wave packets the leading order tunneling component is gaussian for sufficiently small we prove this for both the large time asymptotics and for moderately large values of the time variable

waves near stokes lines the large k asymptotics of d 2 u z d z 2 k 2 r 2 z u z are studied near a stokes line z z 0 r d z real where z 0 is a zero of r 2 z of any order on which there is greatest disparity between the dominant and subdominant exponential waves in the phase integral wkb approximations the aim is to establish precisely how the multiplier b of the subdominant wave varies across the stokes line although b always has a total change proportional to i times the multiplier of the dominant wave the stokes phenomenon the form of the change depends on the convention used to define the two waves the optimal convention for which the variation is maximally compact and smooth is to define them by the phase integral approximation truncated at its least term whose order is proportional to k and therefore large asymptotics of asymptotics then the variation of b is proportional to the error function of the natural stokes crossing variable im k re this result is obtained without resumming divergent series thereby avoiding asymptotics of asymptotics of asymptotics an application is given to the birth of exponentially weak reflected waves in media with smoothly varying refractive index

high orders of the weyl expansion for quantum billiards resurgence of periodic orbits and the stokes phenomenon a formalism is developed for calculating high coefficients c r of the weyl high energy expansion for the trace of the resolvent of the laplace operator in a domain b with smooth boundary b the c r are used to test the following conjectures a the sequence of c r diverges factorially controlled by the shortest accessible real or complex periodic geodesic b if this is a 2 bounce orbit it corresponds to the saddle of the chord length function whose contour is first crossed when climbing from the diagonal of the mobius strip which is the space of chords of b c this orbit gives an exponential contribution to the remainder when the weyl series truncated at its least term is subtracted from the resolvent the exponential switches on smoothly according to an error function where it is smallest that is across the negative energy axis stokes line these conjectures are motivated by recent results in asymptotics they survive tests for the circle billiard and for a family of curves with 2 and 3 bulges where the dominant orbit is not always the shortest and is sometimes complex for some systems which are not smooth billiards e g a particle on a ring or in a billiard where b is a polygon the weyl series terminates and so no geodesics are accessible for a particle on a compact surface of constant negative curvature only the complex geodesics are accessible from the weyl series

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

unfolding the high orders of asymptotic expansions with coalescing saddles singularity theory crossover and duality we study the leading behaviour of the late coefficients high orders r of asymptotic expansions in a large parameter k for contour integrals involving a cluster of coalescing saddles and thereby establish the form of the divergence of the expansions the two principal cases are saddle to cluster where the integral is through a simple saddle and its expansion diverges because of a distant cluster and cluster to saddle where the integral is through a cluster and its expansion diverges because of a distant simple saddle in both the large r coefficients are dominated by the factorial divided by power familiar in asymptotics but this changes its form as the saddles in the cluster are made to coalesce and separate by varying parameters a a 1 a 2 in the integrand the crossover between different forms is described by a series of canonical integrals built from the cuspoid catastrophe polynomials of singularity theory that describe the geometry of the coalescence the arguments of these integrals involve not only the a but also fractional powers of r which by a curious duality replace the powers of the original large parameter k which occur in uniform approximations involving these integrals a by product of the cluster to saddle analysis is a new exact formula for the coefficients of uniform asymptotic expansions

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

bosonic realization of toroidal lie algebras of classical types we use fermionic operators to construct toroidal lie algebras of classical types including in particular symplectic affine algebras which are first realized by fermions

universal transition prefactors derived by superadiabatic renormalization for time dependent two state quantum systems the transition probability is exponentially small in the adiabatic parameter in with the exponent determined by a transition point tc in the complex time plane the authors study the in independent prefactors associated with different sorts of transition point which need not correspond to complex degeneracies of the adiabatic energy unlike previous approaches the method they use does not make use of special functions it consists of applying first order perturbation theory to the schrodinger equation obtained by transforming to a series of superadiabatic bases clinging ever more closely to the evolving state if the original matrix elements share a leading singularity t tc r and their fractional deviation from this is t tc s the prefactor is 4 sin2 pi s 2 2r s 2 this is universal in the sense of being invariant under time reparametrization and quantum changes of frame

irreducible representations of virasoro toroidal lie algebras toroidal lie algebras and their vertex operator representations were introduced in mey and a class of indecomposable modules were investigated in this work we extend the toroidal algebra by the virasoro algebra thus constructing a semi direct product algebra containing the toroidal algebra as an ideal and the virasoro algebra as a subalgebra with the use of vertex operators and certain oscillator representations of the virasoro algebra it is proved that the corresponding fock space gives rise to a class of irreducible modules for the virasoro toroidal algebra

bosonic and fermionic representations of lie algebra central extensions abstract given any representation of an arbitrary lie algebra g over a field k of characteristic 0 we construct representations of a central extension of g on bosonic and fermionic fock space the method gives an explicit formula for a sometimes trivial 2 cocycle in h 2 g k we illustrate these techniques with several concrete examples

infinitely many stokes smoothings in the gamma function the stokes lines for gamma z are the positive and negative imaginary axes where all terms in the divergent asymptotic expansion for ln gamma z have the same phase on crossing these lines from the right to the left half plane infinitely many subdominant exponentials appear rather than the usual one the exponentials increase in magnitude towards the negative real axis anti stokes line where they add to produce the poles of gamma z corresponding to each small exponential is a separate component asymptotic series in the expansion for ln gamma z if each is truncated near its least term its exponential switches on smoothly across the stokes lines according to the universal error function law by appropriate subtractions from ln gamma z the switching on of successively smaller exponentials can be revealed the procedure is illustrated by numerical computations

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

slow non hermitian cycling exact solutions and the stokes phenomenon for non hermitian hamiltonians with an isolated degeneracy exceptional point a model for cycling around loops that enclose or exclude the degeneracy is solved exactly in terms of bessel functions floquet solutions returning exactly to their initial states up to a constant are found as well as exact expressions for the adiabatic multipliers when the evolving states are represented as a superposition of eigenstates of the instantaneous hamiltonian adiabatically i e for slow cycles the multipliers of exponentially subdominant eigenstates can vary wildly unlike those driven by hermitian operators which change little these variations are explained as an example of the stokes phenomenon of asymptotics improved superadiabatic approximations tame the variations of the multipliers but do not eliminate them

propagation through conical crossings an asymptotic semigroup we consider the standard model problem for a conical intersection of electronic surfaces in molecular dynamics our main result is the construction of a semi group in order to approximate the wigner function associated with the solution of the schrodinger equation at leading order in the semiclassical parameter the semigroup stems from an underlying markov process that combines deterministic transport along classical trajectories within the electronic surfaces and random jumps between the surfaces near the crossing our semigroup can be viewed as a rigorous mathematical counterpart of so called trajectory surface hopping algorithms which are of major importance in molecular simulations in chemical physics the key point of our analysis the incorporation of the nonadiabatic transitions is based on the landau zener type formula of fermanian kammerer and gerard10 for the propagation of two scale wigner measures through conical crossings 2005 wiley periodicals inc

vertex representations for n toroidal lie algebras and a generalization of the virasoro algebra vertex representations are obtained for toroidal lie algebras for any number of variables these representations afford representations of certainn variable generalizations of the virasoro algebra that are abelian extensions of the lie algebra of vector fields on a torus

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

a time dependent born oppenheimer approximation with exponentially small error estimates we present the construction of an exponentially accurate time dependent born oppenheimer approximation for molecular quantum mechanics

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

tests for nonergodicity of denumerable continuous time markov processes we provide nonergodicity criteria for denumerable continuous time markov processes in terms of test functions that satisfy kaplan s condition which resolves an open problem given by b d choi b kim nonergodicity criteria for denumerable continuous time markov processes and operations research letters 32 2004 575 580 we give two examples where the nonergodicity criteria are used

acoustic vorticity coupling in linearly varying shear flows using the wkb method the evolution of acoustic and vorticity perturbations in a two dimensional incompressible linear flow is investigated a weighted decomposition of the flow into a hyperbolic part and a rotation par

the riemann siegel expansion for the zeta function high orders and remainders on the critical line s i t t real riemann s zeta function can be calculated with high accuracy by the riemann siegel expansion this is derived here by elementary formal manipulations of the dirichlet series it is shown that the expansion is divergent with the high orders r having the familiar 9factorial9 divided by power9 dependence decorated with an unfamiliar slowly varying multiplier function which is calculated explicitly terms of the series decrease until r r 2 t and then increase the form of the remainder when the expansion is truncated near r is determined it is of order exp t indicating that the critical line is a stokes line for the riemann siegel expansion these conclusions are supported by computations of the first 50 coefficients in the expansion and of the remainders as a function of truncation for several values of t