second order perturbation theory for embedded eigenvalues we study second order perturbation theory for embedded eigenvalues of an abstract class of self adjoint operators using an extension of the mourre theory under assumptions on the regularity of bound states with respect to a conjugate operator we prove upper semicontinuity of the point spectrum and establish the fermi golden rule criterion our results apply to massless pauli fierz hamiltonians for arbitrary coupling

ruin probability with claims modeled by a stationary ergodic stable process for a random walk with negative drift we study the exceedance probability ruin probability of a high threshold the steps of this walk claim sizes constitute a stationary ergodic stable process we study how ruin occurs in this situation and evaluate the asymptotic behavior of the ruin probability for a large variety of stationary ergodic stable processes our findings show that the order of magnitude of the ruin probability varies significantly from one model to another in particular ruin becomes much more likely when the claim sizes exhibit long range dependence the proofs exploit large deviation techniques for sums of dependent stable random variables and the series representation of a stable process as a function of a poisson process

analyticity of resonances and eigenvalues and spectral properties of the massless spin boson model abstract we extend the method of pizzo multiscale analysis for resonances introduced in 5 in order to infer analytic properties of resonances and eigenvalues and their eigenprojections as well as estimates for the localization of the spectrum of dilated hamiltonians and norm bounds for the corresponding resolvent operators in neighborhoods of resonances and eigenvalues we apply our method to the massless spin boson model assuming a slight infrared regularization we prove that the resonance and the ground state eigenvalue and their eigenprojections are analytic with respect to the dilation parameter and the coupling constant moreover we prove that the spectrum of the dilated spin boson hamiltonian in the neighborhood of the resonance and the ground state eigenvalue is localized in two cones in the complex plane with vertices at the location of the resonance and the ground state eigenvalue respectively additionally we provide norm estimates for the resolvent of the dilated spin boson hamiltonian near the resonance and the ground state eigenvalue the topic of analyticity of eigenvalues and resonances has let to several studies and advances in the past however to the best of our knowledge this is the first time that it is addressed from the perspective of pizzo multiscale analysis once the multiscale analysis is set up our method gives easy access to analyticity essentially it amounts to proving it for isolated eigenvalues only and use that uniform limits of analytic functions are analytic the type of spectral and resolvent estimates that we prove are needed to control the time evolution including the scattering regime the latter will be demonstrated in a forthcoming publication the introduced multiscale method to study spectral and resolvent estimates follows its own inductive scheme and is independent and different from the method we apply to construct resonances

one boson scattering processes in the massive spin boson model abstract the spin boson model describes a two level quantum system that interacts with a second quantized boson scalar field recently the relation between the integral kernel of the scattering matrix and the resonance in this model has been established in 18 for the case of massless bosons in the present work we treat the massive case on the one hand one might rightfully expect that the massive case is easier to handle since in contrast to the massless case the corresponding hamiltonian features a spectral gap on the other hand it turns out that the non zero boson mass introduces a new complication as the spectrum of the complex dilated free hamiltonian exhibits lines of spectrum attached to every multiple of the boson rest mass energy starting from the ground and excited state energies this leads to an absence of decay of the corresponding complex dilated resolvent close to the real line which in 18 was a crucial ingredient to control the time evolution in the scattering regime with the new strategy presented here we provide a proof of an analogous formula for the scattering kernel as compared to the massless case and use the opportunity to provide the required spectral information by a mourre theory argument combined with a suitable application of the feshbach schur map instead of complex dilation

spectral and scattering theory of spatially cut off p ϕ 2 hamiltonians we study spatially cut off p 2 hamiltonians we show the local finiteness of the pure point spectrum outside of thresholds the limiting absorption principle and asymptotic completeness of scattering for such hamiltonians our results imply the absence of singular continuous spectrum

energy critical hartree equation with harmonic potential for radial data abstract in this paper we consider the defocusing energy critical hartree equation with harmonic potential for the radial data in all dimensions n 5 and show the global well posedness and scattering theory in the space h 1 f h 1 we take advantage of some symmetry of the hartree nonlinearity to exploit the derivative like properties of the galilean operators and obtain the energy control as well based on bourgain and tao s approach we use a localized morawetz identity to show the global well posedness a key decay estimate comes from the linear part of the energy rather than the nonlinear part which finally helps us to complete the scattering theory

nonlinear scattering with nonlocal interaction we consider the scattering problem for the hartree type equation in n withn 2 i frac partial u partial t frac 1 2 delta u v u 2 u where v x sum limits j 1 2 lambda j x gamma j lambda 1 lambda 2 ne 0 0 lambda j in mathbb r gamma j 0 and denotes the convolution in n we prove the existence of wave operators inh0 k l2 n x k l2 n for any positive integerk under the assumption 1 1 2 2 this is an optimal result in the sense that the existence of wave operators breaks down if min 1 2 1 the case where 1 1 2 2 is also treated according to the sign of 2

functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows we establish a new class of functional central limit theorems for partial sum of certain symmetric stationary infinitely divisible processes with regularly varying levy measures the limit process is a new class of symmetric stable self similar processes with stationary increments that coincides on a part of its parameter space with a previously described process the normalizing sequence and the limiting process are determined by the ergodic theoretical properties of the flow underlying the integral representation of the process these properties can be interpreted as determining how long the memory of the stationary infinitely divisible process is we also establish functional convergence in a strong distributional sense for conservative pointwise dual ergodic maps preserving an infinite measure

renormalization group analysis of spectral problems in quantum field theory abstract in this paper we present a self contained and detailed exposition of the new renormalization group technique proposed in 1 2 its main feature is that the renormalization group transformation acts directly on a space of operators rather than on objects such as a propagator the partition function or correlation functions we apply this renormalization transformation to a hamiltonian describing the physics of an atom interacting with the quantized electromagnetic field and we prove that excited atomic states turn into resonances when the coupling between electrons and field is nonvanishing

smooth feshbach map and operator theoretic renormalization group methods abstract a new variant of the isospectral feshbach map defined on operators in hilbert space is presented it is constructed with the help of a smooth partition of unity instead of projections and is therefore called smooth feshbach map it is an effective tool in spectral and singular perturbation theory as an illustration of its power a novel operator theoretic renormalization group method is described and applied to analyze a general class of hamiltonians on fock space the main advantage of the new renormalization group method over its predecessors is its technical simplicity which it owes to the use of the smooth feshbach map

volatility activity specification and estimation abstract the paper examines volatility activity and its asymmetry and undertakes further specification analysis of volatility models based on it we develop new nonparametric statistics using high frequency option based vix data to test for asymmetry in volatility jumps we also develop methods for estimating and evaluating using price data alone a general encompassing model for volatility dynamics where volatility activity is unrestricted the nonparametric application to vix data along with model estimation for s p index returns suggests that volatility moves are best captured by an infinite variation pure jump martingale with a symmetric jump compensator around zero the latter provides a parsimonious generalization of the jump diffusions commonly used for volatility modeling

absence of singular continuous spectrum for certain self adjoint operators we give a sufficient condition for a self adjoint operator to have the following properties in a neighborhood of a pointe of its spectrum a its point spectrum is finite b its singular continuous spectrum is empty c its resolvent satisfies a class of a priori estimates

commutators c0 semigroups and resolvent estimates abstract we study the existence and the continuity properties of the boundary values on the real axis of the resolvent of a self adjoint operator h in the framework of the conjugate operator method initiated by mourre we allow the conjugate operator a to be the generator of a c 0 semigroup finer estimates require a to be maximal symmetric and we consider situations where the first commutator h i a is not comparable to h the applications include the spectral theory of zero mass quantum field models

quantum electrodynamics of confined nonrelativistic particles abstract we consider a system of finitely many nonrelativistic quantum mechanical electrons bound to static nuclei the electrons are minimally coupled to the quantized electromagnetic field but we impose an ultraviolet cutoff on the electromagnetic vector potential appearing in covariant derivatives and the interactions between the radiation field and electrons localized very far from the nuclei are turned off for a class of hamiltonians we prove exponential localization of bound states establish the existence of a ground state and derive sufficient conditions for its uniqueness furthermore we show that excited bound states of the unperturbed system become unstable and turn into resonances when the electrons are coupled to the radiation field to this end we develop a novel renormalization transformation which acts directly on the space of hamiltonians

continuous renormalization group analysis of spectral problems in quantum field theory the isospectral renormalization group is a powerful method to analyze the spectrum of operators in quantum field theory it was introduced in 1995 see 2 4 and since then it has been used to prove several results for non relativistic quantum electrodynamics after the introduction of the method there have been many works in which extensions simplifications or clarifications are presented see 7 11 13 in this paper we present a new approach in which we construct a flow of operators parametrized by a continuous variable in the positive real axis while this is in contrast to the discrete iteration used before this is more in spirit of the original formulation of the renormalization group introduced in theoretical physics in 1974 22 the renormalization flow that we construct can be expressed in a simple way it can be viewed as a single application of the feshbach schur map with a clever selection of the spectral parameter another advantage of the method is that there exists a flow function for which the renormalization group that we present is the orbit under this flow of an initial hamiltonian this opens the possibility to study the problem using different techniques coming from the theory of evolution equations

maximal velocity of photons in non relativistic qed abstract we consider the problem of propagation of photons in the quantum theory of non relativistic matter coupled to electromagnetic radiation which is presently the only consistent quantum theory of matter and radiation assuming that the matter system is in a localized state i e for energies below the ionization threshold we show that the probability to find photons at time t at the distance greater than c t where c is the speed of light vanishes as t as an inverse power of t

on the smooth feshbach schur map a new variant of the feshbach map called smooth feshbach map has been introduced recently by bach et al in connection with the renormalization analysis of non relativistic quantum electrodynamics we analyze and clarify its algebraic and analytic properties and we generalize it to non selfadjoint partition operators and

perturbation of embedded eigenvalues in the generalized n body problem we discuss the perturbation of continuum eigenvalues without analyticity assumptions among our results we show that generally a small perturbation removes these eigenvalues in accordance with fermi s golden rule thus generically in a baire category sense the schrodinger operator has no embedded non threshold eigenvalues

asymptotic completeness for the massless spin boson model abstract we consider generalized versions of the massless spin boson model building on the recent work in 3 and 4 we prove asymptotic completeness

positive commutators fermi golden rule and the spectrum of zero temperature pauli fierz hamiltonians abstract we perform the spectral analysis of a zero temperature pauli fierz system for small coupling constants under the hypothesis of fermi golden rule we show that the embedded eigenvalues of the uncoupled system disappear and establish a limiting absorption principle above this level of energy we rely on a positive commutator approach introduced by skibsted and pursued by georgescu gerard moller we complete some results obtained so far by derezinski jaksic on one side and by bach frohlich sigal soffer on the other side

high level excursion set geometry for non gaussian infinitely divisible random fields nsa grant mspf 05g 049 aro grant w911nf 07 1 0078 us israel binational science foundation grant 2004064

estimation of jump tails we propose a new and flexible non parametric framework for estimating the jump tails of ito semimartingale processes the approach is based on a relatively simple to implement set of estimating equations associated with the compensator for the jump measure or its intensity that only utilizes the weak assumption of regular variation in the jump tails along with in fill asymptotic arguments for directly estimating the large jumps the procedure assumes that the large sized jumps are identically distributed but otherwise allows for very general dynamic dependencies in jump occurrences and importantly does not restrict the behavior of the small jumps nor the continuous part of the process and the temporal variation in the stochastic volatility on implementing the new estimation procedure with actual high frequency data for the s p 500 aggregate market portfolio we find strong evidence for richer and more complex dynamic dependencies in the jump tails than hitherto entertained in the literature

existence and construction of resonances for atoms coupled to the quantized radiation field abstract the processes of emission and absorption of photons by atoms can be rigorously understood in the low energy limit if we neglect the creation and annihilation of electrons they are related to eigenvalues of the atomic hamiltonian that are embedded in the continuous spectrum of the free hamiltonian of the electromagnetic field the mathematical analysis of the perturbation of these eigenvalues due to the electromagnetic interaction relies on a complex deformation technique relating the original hamiltonian to a non selfadjoint operator we develop a new technique to analyze the spectrum of operators used in non relativistic quantum electrodynamics our method can be applied to prove most of the results that previously required an involved renormalization group construction we use analytic perturbation theory of operators in hilbert spaces instead more precisely we extend the multi scale analysis introduced by one of us in 2003 15 which was used so far only for the study of selfadjoint operators to non selfadjoint operators compared to the selfadjoint case see for example 3 the analysis of these non selfadjoint operators is more difficult because we cannot make use of the functional calculus spectral theorem and the min max principle in some crucial estimates