20 research outputs found
Gibbs measures on Brownian currents
Gibbs measures on Brownian current
Typical long time behaviour of ground state-transformed jump processes
We consider a class of Levy-type processes derived via a Doob-transform from Levy processes conditioned by a control function called potential. These processes have position-dependent and generally unbounded components, with stationary distributions given by the ground states of the Levy generators perturbed by the potential. We derive precise lower and upper envelopes for the almost sure long time behaviour of these ground state-transformed Levy processes, characterized through escape rates and integral tests. We also highlight the role of the parameters by specific examples
Embedded eigenvalues and Neumann–Wigner potentials for relativistic Schrödinger operators
The existence of potentials for relativistic Schrodinger operators allowing eigenvalues em
bedded in the essential spectrum is a long-standing open problem. We construct Neumann-Wigner type potentials for the massive relativistic Schrodinger operator in one and three dimensions for which an embedded eigenvalue exists. We show that in the non-relativistic limit these potentials converge to the classical Neumann-Wigner and Moses-Tuan potentials, respectively. For the massless operator in one dimension we construct two families of potentials, different by the parities of the (generalized) eigenfunctions, for which an
eigenvalue equal to zero or a zero-resonance exists, dependent on the rate of decay of
the corresponding eigenfunctions. We obtain explicit formulae and observe unusual decay
behaviours due to the non-locality of the operator
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
Zero-energy bound state decay for non-local Schrödinger operators
We consider solutions of the eigenvalue equation at zero energy for a class of non-local Schrödinger operators with potentials decreasing to zero at infinity. Using a path integral approach, we obtain detailed results on the spatial decay at infinity of both L2 and resonance solutions. We highlight the interplay of the kinetic term and the potential in these decay behaviours, and identify the decay mechanisms resulting from specific balances of global lifetimes with or without the potential.<br
Maximum principles for time-fractional Cauchy problems with spatially non-local components
We show a strong maximum principle and an Alexandrov-Bakelman-Pucci estimate for the weak solutions of a Cauchy problem featuring Caputo time-derivatives and non-local operators in space variables given in terms of Bernstein functions of the Laplacian. To achieve this, first we propose a suitable meaning of a weak solution, show their existence and uniqueness, and establish a probabilistic representation in terms of time-changed Brownian motion. As an application, we also discuss an inverse source problem
Ambrosetti-Prodi type results for Dirichlet problems of fractional Laplacian-like operators
We establish Ambrosetti–Prodi type results for viscosity and classical solutions of nonlinear Dirichlet problems for fractional Laplace and comparable operators. In the choice of nonlinearities we consider semi-linear and super-linear growth cases separately. We develop a new technique using a functional integration-based approach, which is more robust in the non-local context than a purely analytic treatment
Spectral properties of the massless relativistic harmonic oscillator
Spectral properties of the massless relativistic harmonic oscillato
The spectrum of non-local discrete Schrödinger operators with a δ-potential
The behaviour of the spectral edges (embedded eigenvalues and resonances) is discussed at the two ends of the
continuous spectrum of non-local discrete Schrödinger operators with a δ-potential. These operators arise by
replacing the discrete Laplacian by a strictly increasing C1-function of the discrete Laplacian. The dependence of the
results on this function and the lattice dimension are explicitly derived. It is found that while in the case of the discrete
Schrödinger operator these behaviours are the same no matter which end of the continuous spectrum is considered,
an asymmetry occurs for the non-local cases. A classification with respect to the spectral edge behaviour is also offered
Ultraviolet renormalization of the Nelson Hamiltonian through functional integration
© 2014 Elsevier Inc. Starting from the N-particle Nelson Hamiltonian defined by imposing an ultraviolet cutoff, we perform ultraviolet renormalization by showing that in the ultraviolet cutoff limit a self-adjoint operator exists after a logarithmically divergent term is subtracted from the original Hamiltonian. We obtain this term as the diagonal part of a pair interaction appearing in the density of a Gibbs measure derived from the Feynman-Kac representation of the Hamiltonian. Also, we show existence of a weak coupling limit of the renormalized Hamiltonian and derive an effective Yukawa interaction potential between the particles