lowest energy states in nonrelativistic qed atoms and ions in motion within the framework of nonrelativisitic quantum electrodynamics we consider a single nucleus and n electrons coupled to the radiation field since the total momentum p is conserved the hamiltonian h admits a fiber decomposition with respect to p with fiber hamiltonian h p a stable atom resp ion means that the fiber hamiltonian h p has an eigenvalue at the bottom of its spectrum we establish the existence of a ground state for h p under i an explicit bound on p ii a binding condition and iii an energy inequality the binding condition is proven to hold for a heavy nucleus and the energy inequality for spinless electrons

an expression of the ground state energy of the spin boson model abstract an expression of the ground state energyesbof the spin boson hamiltonianhsbis considered the expression in the cases of both massive and massless bosons is given by a nonperturbative method by using the expression we show a necessary and sufficient condition with respect to a parameterg 1 0 such that a formula withgattains toesb

a central limit theorem for gibbs measures relative to brownian motion we study a gibbs measure over brownian motion with a pair potential which depends only on the increments assuming a particular form of this pair potential we establish that in the infinite volume limit the gibbs measure can be viewed as brownian motion moving in a dynamic random environment thereby we are in a position to use the technique of kipnis and varadhan and to prove a functional central limit theorem

spectral analysis of coupled linear complementarity problems abstract this note deals with the so called cone constrained bivariate eigenvalue problem the equilibrium model under consideration is a system of linear complementarity problems p x ax by x p q y cx dy y q involving two closed convex cones and their corresponding duals we study the set of pairs r 2 for which this system has a nontrivial solution x y r n m we discuss also the link between the cone constrained version and the unconstrained one

renormalization group and analyticity in one dimension a proof of dobrushin s theorem we consider one dimensional systems described by many body potentials with finite first moment and prove that the correlation functions are analytic in the interaction parameters this result is not new dobrushin 1973 but our proof is simpler and physically more transparent we show that by introducing suitable blocks and averaging over the variables associated to a subset of the blocks decimation procedure the resulting effective interaction is such that the system can always be dealt with as a high temperature system

kronecker bases for linear matrix equations with application to two parameter eigenvalue problems abstract the general solutions of the homogeneous matrix equation axct bxdt 0 and the system of the matrix equations ax by 0 xct ydt 0 are described in terms of kronecker canonical forms i e in terms of kronecker invariants and kronecker bases for pairs of matrices a b and c d a canonical form for a pair of commuting matrices e f such that e2 f2 ef 0 is discussed these results are applied to construct a canonical basis for the second root subspace of a two parameter eigenvalue problem the corresponding relations for canonical invariants are given

functional integral approach to semi relativistic pauli fierz models abstract by means of functional integrations spectral properties of semi relativistic pauli fierz hamiltonians h p a 2 m 2 m v h rad in quantum electrodynamics are considered here p is the momentum operator a a quantized radiation field on which an ultraviolet cutoff is imposed v an external potential h rad the free field hamiltonian and m 0 describes the mass of electron two self adjoint extensions of a semi relativistic pauli fierz hamiltonian are defined the feynman kac type formula of e t h is given a self adjointness a spatial decay of bound states a gaussian domination of the ground state and the existence of a measure associated with the ground state are shown all the results are independent of values of coupling constant and it is emphasized that m 0 is included

a probabilistic representation of the ground state expectation of fractional powers of the boson number operator abstract we give a formula in terms of a joint gibbs measure on brownian paths and the measure of a random time poisson process of the ground state expectations of fractional in fact any real powers of the boson number operator in the nelson model we use this representation to obtain tight two sided bounds as applications we discuss the polaron and translation invariant nelson models

cantor spectrum for the almost mathieu equation for a dense g of pairs in r2 we prove that the operator hu n u n 1 u n 1 cos 2 n u n has a nowhere dense spectrum along the way we prove several interesting results about the case pq of which we mention a if q is not an integral multiple of then all gaps are open and b if q is even and is chosen suitably then the middle gap is closed for all

two weight hilbert transform and lipschitz property of jacobi matrices associated to hyperbolic polynomials abstract we are going to prove a lipschitz property of jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical perron frobenius ruelle operators associated to hyperbolic polynomial dynamics this lipschitz estimate will not depend on the dimension of the jacobi matrix it is obtained using some sufficient conditions for two weight boundedness of the hilbert transform it has been proved in f peherstorfer a volberg p yuditskii limit periodic jacobi matrices with prescribed p adic hull and a singular continuous spectrum math res lett 13 2 3 2006 215 230 for all polynomials with sufficiently big hyperbolicity and in the most symmetric case t 0 that the lipschitz estimate becomes exponentially better when the dimension of the jacobi matrix grows this allows us to get for such polynomials the solution of a problem of bellissard in other words to prove the limit periodicity of the limit jacobi matrix we suggest a scheme how to approach bellissard s problem for all hyperbolic dynamics by uniting the methods of the present paper and those of f peherstorfer a volberg p yuditskii limit periodic jacobi matrices with prescribed p adic hull and a singular continuous spectrum math res lett 13 2 3 2006 215 230 on the other hand the nearness of jacobi matrices under consideration in operator norm implies a certain nearness of their canonical spectral measures one can notice that this last claim just gives us the classical commutative perron frobenius ruelle theorem it is concerned exactly with the nearness of such measures in particular in many situations we can see that the classical perron frobenius ruelle theorem is a corollary of a certain non commutative observation concerning the quantitative nearness of pertinent jacobi matrices in operator norm

fiber hamiltonians in non relativistic quantum electrodynamics abstract a translation invariant hamiltonian h of non relativistic quantum electrodynamics is studied this hamiltonian is decomposed with respect to the total momentum p t h r d h p d p where the self adjoint fiber hamiltonian h p is defined for arbitrary values of coupling constants the relationship between rotation invariance of h p and polarization vectors is discussed and functional integral representations of n point euclidean green functions of h p are given from these applications into the degeneracy of ground states ground state energy and expectation values of suitable observables with respect to ground states are given

on enhanced binding and related effectsin the non and semi relativistic pauli fierz models we prove enhanced binding and increase of binding energies in the non and semi relativistic pauli fierz models for arbitrary values of the fine structure constant and the ultra violet cut off and discuss the resulting improvement of exponential localization of ground state eigenvectors for the semi relativistic model we also discuss the increase of the renormalized electron mass and determine the linear leading order term in the asymptotics of the self energy as the ultra violet cut off goes to infinity

a quantum crystal with multidimensional anharmonic oscillators the quantum anharmonic crystal is made up of a large number of multidimensional anharmonic oscillators arranged in a periodic spatial lattice with a nearest neighbor coupling if the coupling coefficient is sufficiently small then there is a convergent expansion for the ground state of the crystal the estimates on the convergence are independent of the size of the crystal the proof uses the path integral representation of the ground state in terms of diffusion processes the convergence of the cluster expansion depends on the ergodicity properties of these processes

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

gibbs measures for brownian paths under the effect of an external and a small pair potential we consider brownian motion in the presence of an external and a weakly coupled pair interaction potential and show that its stationary measure is a gibbs measure uniqueness of the gibbs measure for two cases is shown also the typical path behaviour the degree of mixing and some further properties are derived we use cluster expansion in the small coupling parameter

a two weight inequality for the hilbert transform assuming an energy hypothesis let and be locally finite positive borel measures on r subject to the pair of weights satisfying a side condition we characterize boundedness of the hilbert transform h from l2 to l2 in terms of the a2 condition i i i x xi 2d x i i i x xi 2d x 12 c i and the two testing conditions for all intervals i in r ih 1i x 2d x c id x ih 1i x 2d x c id x the proof uses the beautiful corona argument of nazarov treil and volberg there is a range of side conditions termed energy conditions at one endpoint the energy conditions are also a consequence of the testing conditions above and at the other endpoint they are the pivotal conditions of nazarov treil and volberg we detail an example which shows that the pivotal conditions are not necessary for boundedness of the hilbert transform

positivity improving operators and hypercontractivity here 1 1 is the famous nelson hypercontractivity inequality 14 which has proved to be extremely useful in euclidean quantum field theory see e g 16 like 1 1 the estimate 1 2 follows very easily from the it6 calculus of brownian motion 3 here in order to make the paper self contained and to introduce the reader to some general principles of interest we submit a new proof of 1 2 by mimicing a standard proof of 1 1 we should like to add that 1 2 has at least a formal resemblance to the familiar correlation inequalities from statistical mechanics see e g 11 12 16 using 1 1 and 1 2 it is simple to obtain similar estimates for the biquantization of a linear contraction on hilbert space sect 4 now suppose a is a self adjoint operator on l 2 of a probability space and set a a maxspec a sup spec a maxspec a where spec a denotes the spectrum of a below we will state conditions that guarantee that the gap a a is proper in fact a simplified version of our main result theorem 5 3 gives a positive numerical lower bound of a a if i a is positivity preserving ii sup ilzf i1 i f 12 1 k 2 and iii inf izfllp v i l f l l 1 f 0 k 0 some 0 p 1 here i iii imply that a has a unique ground state it shall be empha

stochastic differential equations for models of non relativistic matter interacting with quantized radiation fields we discuss hilbert space valued stochastic differential equations associated with the heat semi groups of the standard model of non relativistic quantum electrodynamics and of corresponding fiber hamiltonians for translation invariant systems in particular we prove the existence of a stochastic flow satisfying the strong markov property and the feller property to this end we employ an explicit solution ansatz in the matrix valued case i e if the electron spin is taken into account it is given by a series of operator valued time ordered integrals whose integrands are factorized into annihilation preservation creation and scalar parts the feynman kac formula implied by these results is new in the matrix valued case furthermore we discuss stochastic differential equations and feynman kac representations for an operator valued integral kernel of the semi group as a byproduct we obtain analogous results for nelson s model

an algebraic approach to multiparameter spectral theory root subspaces for multiprameter eigenvalue problems are described using coalgebraic techniques an algorithm is given to construct bases for the root subspaces

essential self adjointness of translation invariant quantum field models for arbitrary coupling constants the hamiltonian of a system of quantum particles minimally coupled to a quantum field is considered for arbitrary coupling constants the hamiltonian has a translation invariant part by means of functional integral representations the existence of an invariant domain under the action of the heat semigroup generated by a self adjoint extension of the translation invariant part is shown with a non perturbative approach it is proved that the hamiltonian is essentially self adjoint on a domain a typical example is the pauli fierz model with spin 1 2 in nonrelativistic quantum electrodynamics for arbitrary coupling constants

geometric aspects of multiparameter spectral theory the paper contains a geometric description of the dimension of the total root subspace of a regular multiparameter system in terms of the intersection multiplicities of its determinantal hypersurfaces the new definition of regularity used is proved to restrict to the familiar definition in the linear case a decomposability problem is also considered it is shown that the joint root subspace of a multiparameter system may be decomposable even when the root subspace of each member is indecomposable

spectral properties of two parameter eigenvalue problems ii we study the self adjoint eigenvalue problem w x 0 in hilbert space for one equation in two parameters here is bounded below with compact resolvent for each 1 2 we give necessary and sufficient conditions for the existence of so that holds with w 0 and we investigate the geometry of the set z 0 of such we also discuss higher order solution sets z i where the ith eigenvalue of w vanishes deriving various asymptotic results in a unified fashion

finite dimensional multiparameter spectral theory the nonderogatory case abstract a completeness theorem for nonderogatory eigenvalues of multiparameter systems is proved in a finite dimensional setting also a basis for the second root subspace of a simple eigenvalue is given weakly elliptic multiparameter systems are introduced it is shown that simple real eigenvalues of such systems are nonderogatory

multiplicity of ground states in quantum field models applications of asymptotic fields abstract ground states of hamiltonian h of quantum field models are investigated the infimum of the spectrum of h is in the edge of its essential spectrum by means of the asymptotic field theory we give a necessary and sufficient condition for that the expectation value of the number operator of ground states is finite from which we give an upper bound of the multiplicity of ground states of h typical examples are massless gsb models and the pauli fierz model with spin 1 2

two parameter eigenvalue problems for matrices abstract the linked pair of two parameter eigenvalue problems t m 1 v m 1 2 v m 2 x m 0 x m 0 x m h m is considered for hermitian matrices t m v m1 v m2 acting on on the finite dimensional hilbert spaces h m m 1 2 we dispense with customary definiteness conditions although for m 1 2 we assume the existence of a linear combination of v m1 v m2 which is definite an embedding technique is used to study eigenvalues r 2 of specified variational index the transition between real and nonreal eigenvalues and algebraic multiplicity

functional integral representations of the pauli fierz model with spin 1 2 a feynman kac type formula for a levy and an infinite dimensional gaussian random process associated with a quantized radiation field is derived in particular a functional integral representation of e thpf generated by the pauli fierz hamiltonian with spin 1 2 in non relativistic quantum electrodynamics is constructed when no external potential is applied hpf turns translation invariant and it is decomposed as a direct integral hpf r3 hpf p dp the functional integral representation of e thpf p is also given although all these hamiltonians include spin nevertheless the kernels obtained for the path measures are scalar rather than matrix expressions as an application of the functional integral representations energy comparison inequalities are derived