József Lőrinczi

GROUND STATE PROPERTIES OF THE NELSON HAMILTONIAN: A GIBBS MEASURE-BASED APPROACH

The Nelson model describes a quantum particle coupled to a scalar Bose field. We study properties of its ground state through functional integration techniques in case the particle is confined by an external potential. We obtain bounds on the average and the variance of the Bose field both in position and momentum space, on the distribution of the number of bosons, and on the position space distribution of the particle.

PATH INTEGRAL REPRESENTATION FOR SCHRÖDINGER OPERATORS WITH BERNSTEIN FUNCTIONS OF THE LAPLACIAN

Path integral representations for generalized Schrodinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Levy subordinators is used, thereby the role of Brownian motion entering the standard Feynman–Kac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrodinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an Lp-Lq bound of the associated generalized Schrodinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

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.

Infrared Regular Representation of the Three-Dimensional Massless Nelson Model

We prove that in the Euclidean representation of the three-dimensional massless Nelson model, the t = 0 projection of the interacting measure is absolutely continuous with respect to a Gaussian measure with a suitably adjusted mean. We also determine the Hamiltonian in the Fock space over this Gaussian measure space.

Fall-Off of Eigenfunctions for Non-Local Schrödinger Operators with Decaying Potentials

We study the spatial decay of eigenfunctions of non-local Schrödinger operators whose kinetic terms are generators of symmetric jump-paring Lévy processes with Kato-class potentials decaying at infinity. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps. We find that the decay rates of eigenfunctions depend on the process via specific preference rates in particular jump scenarios, and depend on the potential through the distance of the corresponding eigenvalue from the edge of the continuous spectrum. We prove that the conditions of the jump-paring class imply that for all eigenvalues the corresponding positive eigenfunctions decay at most as rapidly as the Lévy intensity. This condition is sharp in the sense that if the jump-paring property fails to hold, then eigenfunction decay becomes slower than the decay of the Lévy intensity. We furthermore prove that under reasonable conditions the Lévy intensity also governs the upper bounds of eigenfunctions, and ground states are comparable with it, i.e., two-sided bounds hold. As an interesting consequence, we identify a sharp regime change in the decay of eigenfunctions as the Lévy intensity is varied from sub-exponential to exponential order, and dependent on the location of the eigenvalue, in the sense that through the transition Lévy intensity-driven decay becomes slower than the rate of decay of the Lévy intensity. Our approach is based on path integration and probabilistic potential theory techniques, and all results are also illustrated by specific examples.

Gibbs Measures on Brownian Paths: Theory and Applications

We review our investigations on Gibbs measures relative to Brownian motion, in particular the existence of such measures and their path properties, uniqueness, resp. non-uniqueness. For the case when the energy only depends on increments, we present a functional central limit theorem. We also explain connections with other work and state open problems of interest.

Transition in the decay rates of stationary distributions of Levy motion in an energy landscape

The time evolution of random variables with Lévy statistics has the ability to develop jumps, displaying very different behaviors from continuously fluctuating cases. Such patterns appear in an ever broadening range of examples including random lasers, non-Gaussian kinetics, or foraging strategies. The penalizing or reinforcing effect of the environment, however, has been little explored so far. We report a new phenomenon which manifests as a qualitative transition in the spatial decay behavior of the stationary measure of a jump process under an external potential, occurring on a combined change in the characteristics of the process and the lowest eigenvalue resulting from the effect of the potential. This also provides insight into the fundamental question of what is the mechanism of the spatial decay of a ground state.

Fractional P(ϕ)1-processes and Gibbs measures

We define and prove existence of fractional P(ϕ)1-processes as random processes generated by fractional Schrodinger semigroups with Kato-decomposable potentials. Also, we show that the measure of such a process is a Gibbs measure with respect to the same potential. We give conditions of its uniqueness and characterize its support relating this with intrinsic ultracontractivity properties of the semigroup and the fall-off of the ground state. To achieve that we establish and analyse these properties first.

Gibbs Measures on Brownian Paths

This is a summary of results based on recent work outlining how Gibbs measures can be defined on Brownian paths and what are their most important properties. Such Gibbs measures have a number of applications in Euclidean quantum field theory, statistical mechanics, stochastic (partial) differential equations and other areas.

P(φ)1-process for the spin-boson model and a functional central limit theorem for associated additive functionals

Abstract We construct a random process with stationary increments associated to the Hamiltonian of the spin-boson model consisting of a component describing the spin and a component given by a Schwartz distribution-valued Ornstein-Uhlenbeck process describing the boson field. We use a path integral representation of the Hamiltonian to prove a functional central limit theorem for additive functionals, and derive explicit expressions of the diffusion constant for specific functionals.