intrinsic ultracontractivity and small perturbation for one dimensional generalized diffusion operators abstract we consider a one dimensional generalized diffusion operator g on an open interval which is not necessarily bounded we characterize sp i e 1 is a small perturbation of the operator g in terms of the classification of the boundary points from the view point of diffusion processes and give a sufficient condition for iu i e the associated heat kernel is intrinsically ultracontractive

intrinsic ultracontractivity and eigenfunction estimates for schrodinger operators let d be a domain in rn n 2 and let h h0 v where h0 is a divergence form uniformly elliptic operator with dirichlet boundary conditions and v is in the kato class intrinsic ultracontractivity is proved for the semigroup of h when d is holder domain of order 0 or a uniformly holder domain of order for 0 2 for every 2 there exists a uniformly holder domain of order for which the dirichlet laplacian is not intrinsically ultracontractive for a large class of domains it is shown that the heat kernels for h0 and h decay at the same rate at the boundary applications are given to the lifetime of conditioned brownian motion some of our results seem to be new even for smooth domains

on the toda and kac van moerbeke systems given a solution of the toda lattice we explicitly construct a solution of the kac van moerbeke system related to each other by a miura type transformation as an illustration of our method we derive the n soliton solutions of the kac van moerbeke lattice

critical phenomena in linear elliptic problems abstract in this article we consider the existence and nonexistence of principal eigenvalues for the linear elliptic equation u v x u in r n and u d 1 2 r n where n 3 and v l 1 loc r n with v 0 in r n we establish that when potential v has strong singularities phenomena which appear under critical exponent nonlinearities brezis and nirenberg are present in linear problems as well

dispersive estimates for rational symbols and local well posedness of the nonzero energy nv equation abstract we consider the cauchy problem for the two dimensional novikov veselov equation integrable via the inverse scattering problem for the schrodinger operator with fixed negative energy the associated linear equation is characterized by a rational symbol which is not a polynomial except when the energy parameter is zero with the help of a complex analysis point of view of the problem we establish uniform decay estimates for the linear solution with gain of almost one derivative and we use this result together with fourier decomposition methods and x s b spaces to prove local well posedness in h s s 1 2

dual piecewise analytic bundle shift models of linear operators lettbe a banach space operator with empty point spectrum whose essential spectrum lies on a finite system of possibly intersecting curves under certain conditions ont dual analytic representations oftas a kind of bundle shift and of t as an adjoint bundle shift are constructed a specialization of this scheme is given which allows one to obtain in some cases concrete similarity models oftandt in certain analogues of smirnove2 spaces the latter construction is applied to toeplitz operators with smooth symbols

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

subcriticality and gaugeability of the schrodinger operator we investigate a schrodinger operator 2 v in r d d 3 with a potential v in the class k d satisfying a similar kato condition at infinity and prove an equivalence theorem connecting various conditions on subcriticality strong positivity and gaugeability of the operator

lp norms of non critical schrodinger semigroups we consider schrodinger semigroups e th h v on rn with v c x 2 as x 0 c 12 n 2 2 with h 0 we determine the exact power law divergence of e th p p and of some e th q p as maps from lp to lq the results are expressed most naturally in terms of the power for which there exists a positive resonance such that h 0 x x

a classification of homogeneous operators in the cowen douglas class a complete list of homogeneous operators in the cowen douglas class b n mathbb d is given this classification is obtained from an explicit realization of all the homogeneous hermitian holomorphic vector bundles on the unit disc under the action of the universal covering group of the bi holomorphic automorphism group of the unit disc

heat kernel eigenfunctions and conditioned brownian motion in planar domains abstract some recent results concerning uniform convergence of the shape of the heat kernel to that of the first eigenfunction and the lifetime of h processes in bounded lipschitz domains in r n are extended to all planar domains of finite area

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

lifetimes of conditioned diffusions we investigate when an upper bound on expected lifetimes of conditioned diffusions associated with elliptic operators in divergence and non divergence form can be found the critical value of the parameter is found for each of the following classes of domains l p domains p n 1 uniformly regular twistedl p domains p n 1 and twisted holder domains 1 3 a related parabolic boundary harnack principle is proved

semismall perturbations semi intrinsic ultracontractivity and integral representations of nonnegative solutions for parabolic equations we consider nonnegative solutions of a parabolic equation in a cylinder d i where d is a noncompact domain of a riemannian manifold and i 0 t with 0 t or i 0 under the assumption ssp i e the constant function 1 is a semismall perturbation of the associated elliptic operator on d we establish an integral representation theorem of nonnegative solutions in the case i 0 t any nonnegative solution is represented uniquely by an integral on d 0 md 0 t where md is the martin boundary of d for the elliptic operator and in the case i 0 any nonnegative solution is represented uniquely by the sum of an integral on md 0 and a constant multiple of a particular solution we also show that ssp implies the condition siu i e the associated heat kernel is semi intrinsically ultracontractive

martin compactification of a complete surface with negative curvature in this paper we consider the martin compactification associated with the operator mathcal l delta 1 of a complete non compact surface sigma 2 with negative curvature in particular we investigate positive eigenfunctions with eigenvalue one of the laplace operator delta of sigma 2 and prove a uniqueness result such eigenfunctions are unique up to a positive multiplicative constant if they vanish on the part of the geometric boundary s infty sigma 2 of sigma 2 where the curvature is bounded above by a negative constant and satisfy some growth estimate on the other part of s infty sigma 2 where the curvature approaches zero this uniqueness result plays an essential role in our recent paper cao and he infinitesimal rigidity of steady gradient ricci soliton in dimension three arxiv 1412 2714 2014 preprint in which we prove an infinitesimal rigidity theorem for deformations of certain three dimensional collapsed gradient steady ricci soliton with a non trivial killing vector field

traps for reflected brownian motion consider an open set d r d d 2 and a closed ball b d let e x tb denote the expectation of the hitting time of b for reflected brownian motion in d starting from x d we say that d is a trap domain if supx e x tb a domain d is not a trap domain if and only if the reflecting brownian motion in d is uniformly ergodic we fully characterize the simply connected planar trap domains using a geometric condition and give a number of less complete results for d 2

positive solutions and large time behaviors of schrodinger semigroups simon s problem let h v where v is a multiplication operator by a real valued function v x on rn which is uniformly holder continuous and 1 x 2 2 v x l rn for some 4 the relationship between existence of positive solutions with growth conditions of hg 0 and asymptotic behaviors as t of e th is established using it b simon s problem for h on r2 is solved

martin boundaries of elliptic skew products semismall perturbations and fundamental solutions of parabolic equations abstract we consider positive solutions of elliptic partial differential equations on non compact domains of riemannian manifolds we explicitly determine martin boundaries and martin kernels for a class of elliptic equations in skew product form by exploiting and developing perturbation theory for elliptic equations and short long time estimates for fundamental solutions of parabolic equations

on the double commutant of cowen douglas operators abstract let t be a cowen douglas operator in this paper we study the von neumann algebra v t consisting of operators commuting with both t and t from a geometric viewpoint we identify operators in v t with connection preserving bundle maps on e t the holomorphic hermitian vector bundle associated to t by studying such bundle maps the structure of v t as well as information on reducing subspaces of t can be determined

constructing solutions of the mkdv equation abstract using commutation methods i e n 1 supersymmetry underlying miura s transformation an explicit construction of solutions of the modified korteweg de vries equation given a solution of the ordinary korteweg de vries equation is provided

pointwise eigenfunction estimates and intrinsic ultracontractivity type properties of feynman kac semigroups for a class of levy processes we introduce a class of l evy processes subject to specific regularity conditions and consider their feynman kac semigroups given under a kato class potential using new techniques first we analyze the rate of decay of eigenfunctions at infinity we prove bounds on subaveraging functions from which we derive two sided sharp pointwise estimates on the ground state and obtain upper bounds on all other eigenfunctions next by using these results we analyze intrinsic ultracontractivity and related properties of the semigroup refining them by the concept of ground state domination and asymptotic versions we establish the relationships of these properties derive sharp necessary and sufficient conditions for their validity in terms of the behavior of the l evy density and the potential at infinity define the concept of borderline potential for the asymptotic properties and give probabilistic and variational characterizations these results are amply illustrated by key examples

characterization of intrinsic ultracontractivity for one dimensional schrodinger operators abstract we consider a schrodinger operator l d 2 d x 2 v x on r where v is a real valued measurable function and give an explicit and simple characterization of intrinsic ultracontractivity iu of the schrodinger semigroup generated by l for a wide class of potentials by making use of it we also give new examples of potentials for which the semigroups satisfy iu or non iu

the least squares estimation of time delay between two signals with unknown relative phase shift this correspondence addresses the problem of time delay estimation between two signals with an unknown constant relative phase shift the method of time delay estimation for the pure delay case of chan et al 1 is applied here to derive the least squares estimator lse an explicit expression for the lse of the delay is obtained which is computationally less complex than the maximum likelihood estimator mle 2 it is shown that the lse is unbiased and that the mean square error or the lse is asymptotically equivalent to that of the pertinent mle

on schrodinger operators with multipolar inverse square potentials abstract positivity essential self adjointness and spectral properties of a class of schrodinger operators with multipolar inverse square potentials are discussed in particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of poles ensuring the positivity of the associated quadratic form is established

an explicit estimate of time delay between two signals with an unknown relative phase shift this correspondence presents an explicit method for evaluating the time delay between two signals with a random relative phase shift the analysis applies for gaussian narrow band signals the method is derived from the ml estimator 1 but is much faster than the ml estimator since a search for a maximum is replaced by an explicit formula

a dual geometric theory for bundle shifts abstract inspired by analytic model theory for hilbert space operators and some recent developments in cowen douglas operators we formulate in this paper a geometric theory for bundle shifts and relate it in duality to the geometric theory of cowen douglas operators

the heat kernel of a schrodinger operator with inverse square potential we consider the schrodinger operator h v x with radial potential v which may have singularity at 0 and a quadratic decay at infinity first we study the structure of positive harmonic functions of h and give their precise behavior second under quite general conditions we prove an upper bound for the correspond heat kernel p x y t of the type 0 p x y t ct n 2u min x t u min y t u t 2exp x y 2ct for all x y rn and t 0 where u is a positive harmonic function of h third if u2 is an a2 weight on rn then we prove a lower bound of a similar type

the estimate of time delay between two signals with random relative phase shift this paper presents the maximum likelihood estimate of the time delay between two signals with random relative phase shift the analysis applies for narrow band signals and incorporates a parametric representation of the spread of the phase variation which allows the examination of all cases ranging from uniformly distributed phase shift to known phase shift a realization of the ml estimator is presented which illustrates the difference between optimum processing for random phase shift and pure delay the cramer rao bound is also derived and the performance degradation with respect to the pure delay case is discussed

criticality theory for schrodinger operators with singular potential abstract this paper extends a weighted spectral gap classification for schrodinger operators v to a large class of locally integrable potentials v called balanced potentials here harnack s inequality will not hold in general for such potentials in addition to the standard potentials in the space l l o c p for some p n 2 this class contains potentials for which v is locally bounded with no restriction on v or more generally those for which the positive and negative singularities are separated as well as those for which the interaction of the positive and negative singularities is not too strong we also establish for such potentials under the additional assumption that v l l o c p for some p n 2 versions of the agmon allegretto piepenbrink principle characterising the subcritical critical operators in terms of the cone of positive distributional super solutions

on positivity criticality and the spectral radius of the shuttle operator for elliptic operators in this paper we study the shuttle operator for a second order linear elliptic operator p on a noncompact manifold x zhao has introduced and studied the shuttle operator and its relation to the theory of positive solutions in the case of small perturbations of the laplacian in ir n 3 zhao was motivated by works of chung and varadhan which consider one dimensional schrodinger operators the main purpose of the paper is to extend the above studies we prove that in the general case the spectral radius of the shuttle operator is strictly less equal or strictly greater than 1 if and only if the operator p is respectively subcritical critical or supercritical in x we demonstrate the usefulness of the characterization for proving theorems our approach is purely analytic this research was partially supported by the israel science foundation administered by the israel academy of sciences and humanities and by the fund for the promotion of research at the technion

radial solutions of a semilinear elliptic problem we analyse the set of nonnegative global and radial solutions radial solutions for short of the equation u u p f in r n n 1 where 0 where graphics when f r c r2p 1 p h r with h r o r2p 1 p as r radial solutions continue to exist if h r is sufficiently small at infinity existence however breaks down if h r 0 graphics whenever they exist radial solutions are characterised in terms of their asymptotic behaviour as r

on families of optimal hardy weights for linear second order elliptic operators abstract we construct families of optimal hardy weights for a subcritical linear second order elliptic operator using a one dimensional reduction more precisely we first characterize all optimal hardy weights with respect to one dimensional subcritical sturm liouville operators on a b a b and then apply this result to obtain families of optimal hardy inequalities for general linear second order elliptic operators in higher dimensions as an application we prove a new rellich inequality

quantum scattering at low energies for a class of negative slowly decaying potentials including v x x with 0 2 we study the quantum mechanical scattering theory in the low energy regime using appropriate modifiers of the isozaki kitada type we show that scattering theory is well behaved on the whole continuous spectrum of the hamiltonian including the energy 0 we show that the modified scattering matrices s are welldefined and strongly continuous down to the zero energy threshold similarly we prove that the modified wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces these results are used to derive oscillatory asymptotics of the standard short range and dollard type s matrices for the subclasses of potentials where both kinds of s matrices are defined for potentials whose leading part is x we show that the location of singularities of the kernel of s experiences an abrupt change from passing from positive energies to the limiting energy 0 this change corresponds to the behaviour of the classical orbits under stronger conditions one can extract the leading term of the asymptotics of the kernel of s at its singularities

a probabilistic approach to bounded positive solutions for schrodinger operators with certain classes of potentials consider the equation 1 2 u vu 0 in r d for d 3 for certain classes of potentials v we use probabilistic tools to study the bounded solutions and the positive solutions for a primary motivation is to offer probabilistic intuition for the results

dirichlet parabolicity and l1 liouville property under localized geometric conditions abstract we shed a new light on the l 1 liouville property for positive superharmonic functions by providing many evidences that its validity relies on geometric conditions localized on large enough portions of the space we also present examples in any dimension showing that the l 1 liouville property is strictly weaker than the stochastic completeness of the manifold the main tool in our investigations is represented by the potential theory of a manifold with boundary subject to dirichlet boundary conditions the paper incorporates under a unifying viewpoint some old and new aspects of the theory with a special emphasis on global maximum principles and on the role of the dirichlet green s kernel

similarity classification of holomorphic curves abstract let h be a complex separable hilbert space for an open connected subset of c we shall say that a map f gr n h is a holomorphic curve if there exist n holomorphic h valued functions 1 2 n on such that f 1 n where gr n h denotes the grassmann manifold the set of all n dimensional subspaces of h in the paper we give a similarity classification of some holomorphic curves by using the k group of its commutant algebra as an invariant

large time behavior of the lp norm of schrodinger semigroups abstract letv 0 v c0 rv with v 3 be such that h 1 2 v 0 but for any e 0 1 2 1 e v is not positive we determine the exact rate of divergence of the norm of e th as a map from l to l a number of related problems are discussed

criticality and ground states for second order elliptic equations abstract let p be a second order elliptic operator in a domain r n suppose that p admits a positive minimal green function g p in it is the main purpose of this paper to prove that if w c is a small perturbation of p in then max w x 0 min w x 0 t u u g p outside a compact set in

maximum and anti maximum principlesand eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations in this paper we discuss some new results concerning perturbation theory for second order elliptic partial differential equations related to positivity properties of such equations we continue the study of some different notions of small perturbations and discuss their relations to comparisons of green s functions refined maximum and anti maximum principles ground state and the decay of eigenfunctions in particular we show that if v is a positive function which is a semismall perturbation of a subcritical schrodinger operator h defined on a domain omega subset mathbb r d and phi k k geq 0 are the dirichlet eigenfunctions of the equation hu lambda vu then for any k geq 0 the function phi k phi 0 is bounded and has a continuous extension up to the martin boundary of the pair omega h where phi 0 is the ground state of h with a principal eigenvalue lambda 0

sharp condition on the decay of the potential for the absence of a zero energy ground state of the schrodinger equation the authors prove a sharp criterion on the decay of the potential of a schrodinger operator on r3 that ensures the absence of a zero energy ground state this condition complements results due to simon and lieb 1981 they also give an improved version of their results which however is still not optimal

structure of positive solutions to δ v u 0 in rn on etudie la structure de toutes les solutions positives d une equation de schrodinger stationnaire ou le potentiel v est suppose etre une fonction a valeur reelle appartenant a l p loc r n avec p n 2 pour n 2 et p 1 pour n 1 on considere une formule de representation explicite pour des solutions positives leurs asymptotes a l infini et le probleme de dirichlet avec une condition aux limites a l infini

integral representations of nonnegative solutions for parabolic equations and elliptic martin boundaries abstract we consider nonnegative solutions of a parabolic equation in a cylinder d 0 t where d is a noncompact domain of a riemannian manifold under the assumption iu i e the associated heat kernel is intrinsically ultracontractive we establish an integral representation theorem any nonnegative solution is represented uniquely by an integral on d 0 m d 0 t where m d is the martin boundary of d for the associated elliptic operator we apply it in a unified way to several concrete examples to explicitly represent nonnegative solutions we also show that iu implies the condition sp i e the constant function 1 is a small perturbation of the elliptic operator on d

a unitary invariant for hyponormal operators abstract several complete unitary invariants for a pure hyponormal operator are described and related to already known invariants among them an operator valued distribution supported by the spectrum has a distinguished position the inverse problems and the relations between these invariants are also discussed parts of this paper were motivated by the recent work of k clancey

asymptotic behavior of elementary solutions of one dimensional generalized diffusion equations for all f e lj dm where dm x is the speed measure as a by product we will also note that 1 4 and 1 5 remain valid for p 1 if l t converges to a positive constant as t oo actually our first interest in this subject was inspired by a physicist masuo suzuki who gave 1 5 for long time tail phenomena in statistical physics 12 and 13 also cf 2 and 7 our results here verify his results rigorously as far