combinatorial approach to feynman path integration combinatorial relations can be used to convert the non relativistic time sliced feynman path integral into perturbation expansions these methods reveal that when the time interval is sliced into n increments each order of perturbation theory sustains an error o 1 square root n in this way the author provides exact path integral results for the following potentials delta function comb finite well tunnelling barrier and a generalized exponential cusp for the tunnelling barrier it is seen how the celebrated 1 reflection factor arises in the limit of infinite barrier height the one dimensional coulomb problem is solved as a limiting case of the exponential cusp in addition for power potentials the author indicates how this path integral approach yields sometimes divergent nevertheless asymptotic perturbation expansions

the stable rank of topological algebras and a problem of r g swan abstract various notions of stable ranks are studied for topological algebras some partial answers to r g swan s problem have two banach or good frechet algebras as in the density theorem in k theory the same stable rank are obtained for example a frechet dense subalgebra a of a c algebra b closed under c functional calculus of self adjoint elements has the same bass stable rank as b

explicit solutions of the time dependent schrodinger equation the darboux transformation is used to generate explicit solutions of the time dependent schrodinger equation in one dimension the potentials are a reflectionless potential and certain asymmetric double well potentials if explicit solutions of a second order ordinary differential equation are known then the darboux transformation generates explicit solutions of a related differential equation deift and trubowitz 1979 lamb 1980 when applied to the time indepen dent schrodinger equation this method gives a new equation with a different potential function the purpose of this paper is to apply the procedure to the time dependent schrodinger equation in one dimension a number of new explicit solutions are thereby obtained given by closed formulae which do not require knowledge of the eigenfunc tions

a representation for fermionic correlation functions let d s a be a gaussian measure on the finitely generated grassmann algebra a given an even w a a we construct an operator r on a such that for all f a a this representation of the schwinger functional iteratively builds up feynman graphs by successively appending lines farther and farther from f it allows the pauli exclusion principle to be implemented quantitatively by a simple application of gram s inequality

nonrelativistic coulomb green s function in parabolic coordinates the nonrelativistic coulomb green s function g r1 r2 k is evaluated by explicit summation over discrete and continuum eigenstates in parabolic coordinates this completes the derivation of meixner who was able to obtain only the r1 0 and r2 limiting forms of the green s function further progress is made possible by an integral representation for a product of two whittaker functions given by buchholz we obtain the closed form for the coulomb green s function previously derived by hostler via an analogous summation in spherical polar coordinates the rutherford scattering limit of the green s function is also demonstrated starting with an integral representation in parabolic coordinates

exact propagator for reflectionless potentials the exact space time propagation for a general reflectionless potential is obtained in closed form involving error functions it is indicated how bound states and transmission coefficients may be recovered from asymptotic behaviour of the propagator a sum rule is derived that shows how the leading terms of the short time form of the propagator can be used rigorously in a feynman path integral formalism

absolutely continuous spectra of quantum tree graphs with weak disorder we consider the laplacian on a rooted metric tree graph with branching number k 2 and random edge lengths given by independent and identically distributed bounded variables our main result is the stability of the absolutely continuous spectrum for weak disorder a useful tool in the discussion is a function which expresses a directional transmission amplitude to infinity and forms a generalization of the weyl titchmarsh function to trees the proof of the main result rests on upper bounds on the range of fluctuations of this quantity in the limit of weak disorder

on the boltzmann equation for the lorentz gas we consider the boltzmann grad limit for the lorentz or wind tree model we prove that if is a fixed configuration of scatterer centers belonging to a set of full measure with respect to the poisson distribution with parameter 0 then the evolution of an initial a c particle density tends in the boltzmann grad limit to the solution of the boltzmann equation for the model as an intermediate step we prove that the process of the free path lengths and impact parameters induced by the lebesgue measure on a small region tends to a limiting independent process

exact constructions of square root helmholtz operator symbols the focusing quadratic profile operator symbols play a pivotal role in both the exact well posed one way reformulation of solving the elliptic helmholtz equation and the construction of the generalized bremmer coupling series the inverse square root and square root helmholtz operator symbols are the initial quantities of interest in both formulations in addition to providing the theoretical framework for the development and implementation of the parabolic equation pe method in wave propagation modeling exact standard left and weyl symbol constructions are presented for both the inverse square root and square root helmholtz operators in the case of the focusing quadratic profile in one transverse spatial dimension extending and ultimately unifying the previously published corresponding results for the defocusing quadratic case j math phys 33 1887 1914 1992 both i spectral modal summation representations and ii contour integral representations exploiting the underlying periodicity of the associated quantum mechanical harmonic oscillator problem are derived and ultimately related through the propagating and nonpropagating contributions to the operator symbol high and low frequency asymptotic operator symbol expansions are given along with the exact symbol representations for the corresponding operator rational approximations which provide the basis for the practical computational realization of the pe method moreover while the focusing quadratic profile is in some respects nonphysical the corresponding helmholtz operator symbols nevertheless establish canonical symbol features for more general profiles containing locally quadratic wells

the short distance behavior of φ 4 3 we consider the 3 4 quantum field theory on a torus and study the short distance behavior we reproduce the standard result that the singularities can be removed by a simple mass renormalization for the resulting model we give anl p bound on the short distance regularity of the correlation functions to obtain these results we develop a systematic treatment of the generating functional for correlations using a renormalization group method incorporating background fields

transfer matrices hyperbolic geometry and absolutely continuous spectrum for some discrete schrodinger operators on graphs abstract we prove the existence of absolutely continuous spectrum for a class of discrete schrodinger operators on tree like graphs we consider potentials whose radial behaviour is subject only to an l bound in the transverse direction the potential must satisfy a condition such as periodicity the graphs we consider include binary trees and graphs obtained from a binary tree by adding edges possibly with weights our methods are motivated by the one dimensional transfer matrix method interpreted as a discrete dynamical system on the hyperbolic plane this is extended to more general graphs leading to a formula for the green s function bounds on the green s function then follow from the contraction properties of the transformations that arise in this generalization the bounds imply the existence of absolutely continuous spectrum

diffusion in a weakly random hamiltonian flow we consider the motion of a particle governed by a weakly random hamiltonian flow we identify temporal and spatial scales on which the particle trajectory converges to a spatial brownian motion the main technical issue in the proof is to obtain error estimates for the convergence of the solution of the stochastic acceleration problem to a momentum diffusion we also apply our results to the system of random geometric acoustics equations and show that the energy density of the acoustic waves undergoes a spatial diffusion

conditionally solvable path integral problems some specific conditionally exactly solvable potentials are discussed within the path integral formalism they generalize the usually known potentials by the incorporation of fractional power behaviour and strongly anharmonic terms we find four different kinds of such potential the first is related to the coulomb potential the second is an anharmonic confinement potential and the third and fourth are related to the manning rosen potential

statistical properties of lorentz gas with periodic configuration of scatterers in our previous paper markov partitions for some classes of dispersed billiards were constructed using these partitions we estimate the decay of velocity auto correlation function and prove the central limit theorem of probability theory and donsker s invariance principle for lorentz gas with periodic configuration of scatterers

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

on the path integral in imaginary lobachevsky space the path integral on the single sheeted hyperboloid i e in d dimensional imaginary lobachevsky space is evaluated a potential problem which we call the kepler problem and the case of a constant magnetic field are also discussed

grad o perturbations of massless gaussian fields we investigate weak perturbations of the continuum massless gaussian measure by a class of approximately local analytic functionals and use our general results to give a new proof that the pressure of the dilute dipole gas is analytic in the activity

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

quantum diffusion of the random schrodinger evolution in the scaling limit ii the recollision diagrams we consider random schrodinger equations on mathbb r d for d 3 with a homogeneous anderson poisson type random potential denote by the coupling constant and t the solution with initial data 0 the space and time variables scale as x sim lambda 2 kappa 2 t sim lambda 2 kappa with 0 0 d we prove that in the limit 0 the expectation of the wigner distribution of t converges weakly to the solution of a heat equation in the space variable x for arbitrary l2 initial data the proof is based on a rigorous analysis of feynman diagrams in the companion paper 10 the analysis of the non repetition diagrams was presented in this paper we complete the proof by estimating the recollision diagrams and showing that the main terms i e the ladder diagrams with renormalized propagator converge to the heat equation

bounds on renormalized feynman graphs we prove two bounds on the value of renormalized euclidean feynman graphs one is a relatively crude but widely applicable bound the other a finer bound applicable to 4 4 like models

kinetic limit for wave propagation in a random medium we study crystal dynamics in the harmonic approximation the atomic masses are weakly disordered in the sense that their deviation from uniformity is of the order e the dispersion relation is assumed to be a morse function and to suppress crossed recollisions we then prove that in the limit e 0 the disorder averaged wigner function on the kinetic scale time and space of order e 1 is governed by a linear boltzmann equation

a mechanical model of brownian motion we consider a dynamical system consisting of one large massive particle and an infinite number of light point particles we prove that the motion of the massive particle is in a suitable limit described by the ornstein uhlenbeck process this extends to three dimensions previous results by holley in one dimension

bounds on completely convergent euclidean feynman graphs letg be a euclidean feynman graph containingl g lines we prove that ifg has massive propagators and does not contain any divergent subgraphs its value is bounded bykl g we also prove the infrared analogue of this bound

construction and borel summability of infrared φ 4 4 by a phase space expansion we construct the thermodynamic limit of the critical massless 4 model in 4 dimensions with an ultraviolet cutoff by means of a partly renormalized phase space expansion this expansion requires in a natural way the introduction of effective or running constants and the infrared asymptotic freedom of the model i e the decay of the running coupling constant plays a crucial role we prove also that the correlation functions of the model are the borel sums of their perturbation expansion

localization lengths and boltzmann limit for the anderson model at small disorders in dimension 3 we prove lower bounds on the localization length of eigenfunctions in the three dimensional anderson model at weak disorders our results are similar to those obtained by schlag shubin and wolff j anal math 88 2002 for dimensions one and two we prove that with probability one most eigenfunctions have localization lengths bounded from below by o left lambda 2 over log 1 over lambda right where is the disorder strength this is achieved by time dependent methods which generalize those developed by erdos and yau commun pure appl math liii 667 753 2003 to the lattice and non gaussian case in addition we show that the macroscopic limit of the corresponding lattice random schrodinger dynamics is governed by a linear boltzmann equation

explicit time dependent schrodinger propagators the authors compute explicitly the time dependent schrodinger and heat propagator for the potentials lambda 2m m 1 cosh2 lambda x a delta function potential several cases of periodic delta function potentials and a delta function potential in a square well they also discuss the semiclassical approximation for the delta function potential

quantum diffusion of the random schrodinger evolution in the scaling limit we consider random schrodinger equations on r d for d 3 with a homogeneous anderson poisson type random potential denote by the coupling constant and psi t the solution with initial data psi 0 the space and time variables scale as x sim lambda 2 varkappa 2 text and t sim lambda 2 varkappa text with 0 varkappa varkappa 0 left d right we prove that in the limit 0 the expectation of the wigner distribution of psi t converges weakly to the solution of a heat equation in the space variable x for arbitrary l 2 initial data

path integrals for potential problems with δ function perturbation the author presents several examples of potential problems with a function perturbation by means of path integrals the idea is to sum a perturbation series expansion resulting in an energy dependent green function g e the energy levels en of the perturbed problem are determined by the equation one dimensional case ig v a a en gamma where g v is the green function of the unperturbed problem gamma is the strength of the potential and a its location in r in d dimensional radial problems with a spherically shaped delta function located at r a this equation changes into ig v l a a en ad 1 where l denotes the angular momentum number

a renormalizable field theory the massive gross neveu model in two dimensions the euclidean massive gross neveu model in two dimensions is just renormalizable and asymptotically free thanks to the pauli principle bare perturbation theory with an ultra violet cut off and the correct ansatz for the bare mass is convergent in a disk whose radius corresponds by asymptotic freedom to a small finite renormalized coupling constant therefore the theory can be fully constructed in a perturbative way it satisfies the o s axioms and is the borel sum of the renormalized perturbation expansion of the model