some potential theory for reflecting brownian motion in holder and lipschitz domains bounds are found on the transition densities and green functions for brownian motion with normal reflection in holder and lipschitz domains for lipschitz domains reflecting brownian motion and boundary local time are constructed a harnack inequality valid up to the boundary is proved a probabilistic solution to the neumann problem is given and the kuramochi boundary is identified

invertible completions of band matrices abstract starting with a hermitian band matrix r we fill it in as we try to complete it to an invertible hermitian matrix f often we can make f invertible even though all but one of the maximal hermitian submatrices in the band may be not invertible this paper is a continuation of the author s recent paper on bordered matrices

lp uniqueness of schrodinger operators and the capacitary positive improving property abstract we prove several lp uniqueness results for schrodinger operators l v by means of the feynman kac formula using the m p capacity theory for general markov semigroups we show that the associated feynman kac semigroup is positive improving in the sense of m p capacity improving the well known one in the sense of measure using that capacitary positive improving property and two new inequalities for generalized ornstein uhlenbeck generators we show the essential self adjointness of the ground state diffusion generator l l 2 associated with two dimensional euclidean quantum fields

conditional gauge with unbounded potential let b be a ball in r d and q be a borel function on it we prove that if mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9gqpe0 rq0dxdd9vqaq jfrvkfhe9pgea0dxdar jb9hs0dxdbpyxe9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaamaaxababagaci 4caiaacwhacaggwbaaleaacawg4bgaeyici4sabmoqayaaraaabeaa ieaakiaa bcadawdrbqaamaalaaabagaaiifaiaadghacaggoagaam yeaiaacmcacagg8baabagaaiifaiaadihacqghsislcawg5bgaaiif amaacaaaleqabagaamizaiabgkhitiaaikdaaaaaaogaa8hiagqaci aa rgacagf5baaleaacawgcbaabeqdcqghrii8aaaa 4fb6 mathop sup limits x in bar b int limits b frac q y x y d 2 dy is sma11 enough then mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9gqpe0 rq0dxdd9vqaq jfrvkfhe9pgea0dxdar jb9hs0dxdbpyxe9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaamaaxababagaci 4caiaacwhacaggwbaaleaadawfqaqaaiaadihacqghiiizcawgcbaa meaacawg6bgaeyici4saeyoaiyraamoqaaqabaaaleqaagqaaogaa8 hiagqaciaa veadaqhaawcbagaa4neaaqaaiaa hhaaagcdawadaqa aigacwgacagg4bgaaiicaiaa bcadawdxbqaaiaa fhacawfoagaa4 heamaabaaaleaacagf0baabeaakiaacmcacawgkbgaamidaawcbaga a8hmaaqaaiaadshadawgaaadbagaamoqaaqabaaaniabguiiydaaki aawufacagldbaacawfgagaa8hpaiaa bcacawfragaa8hiaiabg6hi lcaa 5a6b mathop sup limits mathop x in b limits z in partial b e z x left exp int limits 0 t b q x t dt right infty this paper gives a new proof of one of the two main results by aizenman and simon in 1 by a simple and elementary method a basic theorem in chung and rao 2 is extended to the class of q treated in 1

absorption semigroups their generators and schrodinger semigroups on presente des resultats de perturbations pour des c 0 semi groupes on considere le probleme aux valeurs initiales pour l equation de la chaleur avec absorption avec une solution dans l p r v avec une mauvaise fonction v r r on considere le meme probleme dans l 2 r pour l equation de schrodinger

strong continuity of generalized feynman kac semigroups necessary and sufficient conditions let e d e be a strongly local quasi regular symmetric dirichlet form on l2 e m and xt t 0 px x e the diffusion process associated with e d e for u d e e u has a quasi continuous version u and u xt has fukushima s decomposition u xt u x0 mtu ntu where mtu is the martingale part and ntu is the zero energy part in this paper we study the strong continuity of the generalized feynman kac semigroup defined by ptuf x ex entuf xt t 0 two necessary and sufficient conditions for ptu t 0 to be strongly continuous are obtained by considering the quadratic form qu d e b where qu f f e f f e u f2 for f d e b and the energy measure u of u respectively an example is also given to show that ptu t 0 is strongly continuous when u is not a measure of the kato class but of the hardy class with the constant u e 12 cf definition 4 5

positive semidefinite completions of partial hermitian matrices abstract we classify the ranks of positive semidefinite completions of hermitian band matrices and other partially specified hermitian matrices with chordal graphs and specified main diagonals completing a partially specified matrix means filling in the unspecified entries

uniformly integrable operators and large deviations for markov processes abstract in this paper we introduce and investigate the notion of uniformly integrable operators on l p e its relations to classical compactness and hypercontractivity are exhibited several consequences of this notion are established such as perron frobenius type theorems independence on p of the spectral radius in l p p continuity of spectral radius and especially the existence of spectral gap in the irreducible case we also present some infinitesimal criteria ensuring the uniform integrability of a positive semigroup or of its resolvent for a essentially irreducible markov process we show that the uniform integrability in l p of some type of resolvent associated with the transition semigroup implies the large deviation principle of level 3 with some rate function given by a modified donsker varadhan entropy functional we also prove that the uniform integrability condition becomes even necessary in the symmetric case finally we present several applications of our results to feynman kac semigroups to the thermodynamical limits of grand ensembles and to non symmetric markov processes given by girsanov s formula

uniqueness of generalized schrodinger operators and applications abstract we prove necessary and sufficient conditions on r d r 0 a e so that the operator s 2 1 w1 n has exactly one self adjoint extension on w1 p which generates a sub markovian semigroup on w1 n these conditions are shown to be always fulfilled if d 1 and are verified for a large class of functions if d 1 we also prove an infinite dimensional analogue of this result where r d is replaced by some topological vector space e and the lebesgue measure dx by a gaussian measure the role of is taken by some ornstein uhlenbeck operator with general linear drift and dom s bounded smooth cylinder functions both results are derived as special cases from a general perturbation result we also give the corresponding probabilistic version i e we prove uniqueness for the associated martingale problem we present in some detail applications to the case where is a ground state of a hamiltonian in quantum field theory settling a problem which has been open for some time furthermore we apply our results to prove uniqueness in distribution of the diffusion first constructed by g jona lasinio and p k mitter for the stochastic quantization of f 2 field theory

ranks and inertias of hermitian block toeplitz matrices abstract the rules classifying the connections of the ranks of successive hermitian toeplitz matrices were presented in iohvidov s book hankel and toeplitz matrices and forms this paper generalizes those rules to new rules for successive hermitian block toeplitz matrices t m and t m 1 given a hermitian block toeplitz matrix t m we present the symmetric lattice pentagon of all possible inertias for a successor t m 1 these results are based on our work with nir cohen siam j matrix anal appl 19 1998 freund and huckle showed that hermitian block toeplitz matrices which admit singular extensions rank t m 1 rank t m coincide with those which admit a factorization with krylov matrices this paper extends their list of sufficient conditions

conditional gauge theorem for non local feynman kac transforms feynman kac transforms driven by discontinuous additive functionals are studied in this paper for a large class of markov processes general gauge and conditional gauge theorems are established for such transforms furthermore the l 2 infinitesimal generator of the schrodinger semigroup given by a non local feynman kac transform is determined in terms of its associated bilinear form

a new topological approach to the l uniqueness of operators and the l1 uniqueness of fokker planck equations abstract the usual semigroups of kernels on a polish space e are in general not strongly continuous on l e with respect to the norm topology we introduce a new topology on l e such that they become c 0 semigroups for which we can establish a simplified hille yosida theorem the new topology will allow us to introduce the uniqueness of pre generator on l e which turns out to be equivalent to the l 1 uniqueness of the associated fokker planck equation among many others and it is intimately related with the liouville properties for l 1 harmonic functions the uniqueness of several second order elliptic differential operators in l are studied 1 one dimensional diffusion operators a x f b x f 2 schrodinger operators 1 2 v 3 multi dimensional diffusion generator 1 2

several consequences of an inertia theorem abstract in a recent paper in this journal we extended poincare s inequalities which compare the number of positive eigenvalues of a hermitian matrix with the number for a principal submatrix in this paper we present a plethora of simple consequences of that inertia results this includes 1 a formula for the kernels of certain matrices 2 applications to quadratic programming and 3 a new demonstration that certain real hamiltonian matrices associated with the algebraic riccati equation do not have pure imaginary eigenvalues a formula for the kernels of certain self adjoint operators on infinite dimensional inner product spaces is also presented

perturbation of dirichlet forms lower semiboundedness closability and form cores we study perturbations e e qu of dirichlet forms e on some l2 space l2 m given by quadratic forms q g g d with a signed borel measure whose positive and negative parts are smooth measures with respect to the given dirichlet form we give necessary and sufficient conditions for e to be a lower semibounded closed quadratic form with a unique associated continuous semigroup pt on l2 m we also study the associated resolvent semigroup as well as the relation between e and the largest closed quadratic form which is smaller than e in particular we exhibit form cores

inverse scattering transform for the focusing nonlinear schrodinger equation with nonzero boundary conditions the inverse scattering transform for the focusing nonlinear schrodinger equation with non zero boundary conditions at infinity is presented including the determination of the analyticity of the scattering eigenfunctions the introduction of the appropriate riemann surface and uniformization variable the symmetries discrete spectrum asymptotics trace formulae and the so called theta condition and the formulation of the inverse problem in terms of a riemann hilbert problem in addition the general behavior of the soliton solutions is discussed as well as the reductions to all special cases previously discussed in the literature

uniqueness of generalized schrodinger operators part ii abstract we prove that for h 1 2 loc r d dx 0 dx a e the generalized schrodinger operator s 2 1 dom s c 0 r d has exactly one self adjoint extension on l 2 r d 2 dx which generates a sub markovian semigroup on l 2 r d 2 dx this is based on our previous work where a necessary and sufficient condition on for this to hold was proved but which was only verified to always hold if d 1 we also prove a corresponding result where r d is replaced by an infinite dimensional space and the lebesgue measure by some gaussian measure whose covariance operator has a discrete spectrum

on the connection between schrodinger and dirichlet forms relations between schrodinger forms associated with schrodinger operators in l2 dnx rn open n 1 and the corresponding dirichlet forms are investigated various concrete examples are presented

stochastic calculus on distorted brownian motion relations between malliavin s calculus and the formulation of schrodinger dynamics in terms of local dirichlet forms are investigated it is proved that under mild regularity conditions there exists a directional derivative of the solution of the stochastic equation associated with the corresponding dirichlet form also sufficient conditions are given so that e f xt x is a strong solution of the cauchy problem t h t x q x h t x where xt x t 0 x rd is a distorted brownian motion

non symmetric perturbations of symmetric dirichlet forms abstract we provide a path space integral representation of the semigroup associated with the quadratic form obtained by lower order perturbation of a symmetric local dirichlet form the representation is a combination of feynman kac and girsanov formulas and extends previously known results in the framework of symmetric diffusion processes through the use of the hardy class of smooth measures which contains the kato class of smooth measures

uniqueness of schrodinger operators restricted in a domain abstract let s 2 v be the schrodinger s operator defined on c 0 d where d is a open domain in r d by means of the asymptotic behavior of v near the boundary d we give the necessary and sufficient conditions to the essential markovian self adjointness of s for the nonnegative potential v and to the uniqueness of s in l 1 d for general v

perturbation theory for convolution semigroups abstract we deal with convolution semigroups not necessarily symmetric in l p r n and provide a general perturbation theory of their generators by indefinite singular potentials such semigroups arise in the theory of levy processes and cover many examples such as gaussian semigroups stable semigroups relativistic schrodinger semigroups etc we give new generation theorems and feynman kac formulas in particular by using weak compactness methods in l 1 we enlarge the extended kato class potentials used in the theory of markov processes in l 2 setting dirichlet form perturbation theory is finely related to l 1 theory and the extended kato class measures is also enlarged finally various perturbation problems for subordinate semigroups are considered

chaotic transport and integrable instabilities in a near integrable many particle hamiltonian lattice abstract this paper is a report of detailed numerical experiments on a large dimensional hamiltonian lattice which arises as a nearly integrable discretization of the periodic sine gordon equation through four different one parameter families of initial data we focus on the interplay between numerically observable near integrable behavior quantified by modern statistical diagnostics and the integrable pde solutions and phase space geometry associated with the initial data transitions are observed within each family from quasiperiodic behavior to chaotic motion which is sometimes accompanied by transport toward equipartition these transitions in the near integrable system are correlated with integrable features of the initial data in particular transitions from neutrally stable to exponentially unstable sine gordon solutions and associated homoclinic orbits geometrically these correspond to transitions from neutrally stable tori to whiskered tori our results indicate that 1 positive lyapunov exponents our messure of chaos in near integrable orbits coincide one to one with integrable pde instabilities and homoclinic orbits 2 the statistical transition to equipartition of energy on a long fixed timescale occurs only if but not necessarily if there are integrable instabilities and associated whiskered tori the rate of transport appears to depend on the amplitude of the homoclinic degrees of freedom 3 isolated cases of discreteness induced instabilities and action transport are observed these return to quasiperiodic flow at finer discretizations 4 a simple measure such as energy of the initial data is not an accurate predictor of chaotic or recurrent behavior one family of initial data exhibits as the energy increases alternating windows of quasiperiodic and chaotic orbits

the integrable nature of modulational instability we investigate the nonlinear stage of the modulational or benjamin feir instability by characterizing the initial value problem for the focusing nonlinear schrodinger nls equation with nonzero boundary conditions nzbc at infinity we do so using the recently formulated inverse scattering transform ist for this problem while the linearization of the nls equation ceases to be valid when the perturbations have grown sufficiently large compared to the background the results of the ist remain valid for all times and therefore provide a convenient way to study the nonlinear stage of the modulational instability we begin by studying the spectral problem for the dirac operator i e the first half of the lax pair for the nls equation with piecewise constant initial conditions which are a generalization to nzbc of a potential well and a potential barrier since the scattering data uniquely determine the time evolution of the initial condition via the inverse problem the study of these kinds of poten

inverse scattering transform for the defocusing nonlinear schrodinger equation with fully asymmetric non zero boundary conditions abstract we formulate the inverse scattering transform ist for the defocusing nonlinear schrodinger nls equation with fully asymmetric non zero boundary conditions i e when the limiting values of the solution at space infinities have different non zero moduli the theory is formulated without making use of riemann surfaces and instead by dealing explicitly with the branched nature of the eigenvalues of the associated scattering problem for the direct problem we give explicit single valued definitions of the jost eigenfunctions and scattering coefficients over the whole complex plane and we characterize their discontinuous behavior across the branch cut arising from the square root behavior of the corresponding eigenvalues we pose the inverse problem as a riemann hilbert problem on an open contour and we reduce the problem to a standard set of linear integral equations finally for comparison purposes we present the single sheet branch cut formulation of the inverse scattering transform for the initial value problem with symmetric equimodular non zero boundary conditions as well as for the initial value problem with one sided non zero boundary conditions and we also briefly describe the formulation of the inverse scattering transform when a different choice is made for the location of the branch cuts

nonautonomous kato classes of measures and feynman kac propagators the behavior of the feynman kac propagator corresponding to a time dependent measure on r n is studied we prove the boundedness of the propagator in various function spaces on r n and obtain a uniqueness theorem for an exponentially bounded distributional solution to a nonautonomous heat equation

modulation instability the beginning abstract we discuss the early history of an important field of sturm and drang in modern theory of nonlinear waves it is demonstrated how scientific demand resulted in independent and almost simultaneous publications by many different authors on modulation instability a phenomenon resulting in a variety of nonlinear processes such as envelope solitons envelope shocks freak waves etc examples from water wave hydrodynamics electrodynamics nonlinear optics and convection theory are given

a quasi periodic route to chaos in a near integrable pde abstract pattern formation and transitions to chaos are described for the damped ac driven one dimensional periodic sine gordon equation in a nonlinear schrodinger regime a generic quasi periodic route to intermittent chaos is exhibited in detail using a range of dynamical systems diagnostics in addition a nonlinear spectral transform is exploited i to identify and quantify coordinates of space time attractors in terms of a small number of soliton modes of the underlying integrable system ii to use these analytic coordinates to identify homoclinic orbits as possible sources of chaos and iii to demonstrate the significance of energy transfer between coherent and extended states in this chaotic system

feynman kac semigroups ground state diffusions and large deviations we study the generalized schrodinger operator l v where l is the generator of a symmetric markov semigroup pt on l2 e m and the corresponding dirichlet form ev by means of the cramer functional v we give necessary and sufficient conditions for ev to be lower bounded and for the feynman kac semigroup pvt to be bounded some sufficient conditions for the essential self adjointness of l v are also given by means of large deviations we find a new condition which ensures the existence of ground state of l v and we construct the ground state process q t whose generator is given in the diffusion case by l l 1 where is the square field operator associated to l the self adjointness of l is discussed as applications we consider perturbation of the semigroups of second quantization on an abstract wiener space the time evolution of euclidean quantum fields and stochastic quantization