arXiv subjects

József Lőrinczi

Publications and source records attributed to József Lőrinczi.

18 recordsLinked to original sources

arXiv2025

Special potentials for relativistic Laplacians I: Fractional Rollnik-class

We propose a counterpart of the classical Rollnik-class of potentials for fractional and massive relativistic Laplacians, and describe this space in terms of appropriate Riesz potentials. These definitions rely on precise resolvent estimates. We show that Coulomb-type potentials are elements of fractional Rollnik-class up to but not including the critical singularity of the Hardy potential. For the operators with fractional exponent $α= 1$ there exists no fractional Rollnik potential, however, in low dimensions we make sense of these classes as limiting cases by using $Γ$-convergence. In a second part of the paper we derive detailed results on the self-adjointness and spectral properties of relativistic Schrödinger operators obtained under perturbations by fractional Rollnik potentials. We also define an extended fractional Rollnik-class which is the maximal space for the Hilbert-Schmidt property of the related Birman-Schwinger operators.

math.FA↗

arXiv2022

Bulk behaviour of ground states for relativistic Schrödinger operators with compactly supported potentials

We propose a probabilistic representation of the ground states of massive and massless Schrödinger operators with a potential well in which the behaviour inside the well is described in terms of the moment generating function of the first exit time from the well, and the outside behaviour in terms of the Laplace transform of the first entrance time into the well. This allows an analysis of their behaviour at short to mid-range from the origin. In a first part we derive precise estimates on these two functionals for stable and relativistic stable processes. Next, by combining scaling properties and heat kernel estimates, we derive explicit local rates of the ground states of the given family of non-local Schrödinger operators both inside and outside the well. We also show how this approach extends to fully supported decaying potentials. By an analysis close-by to the edge of the potential well, we furthermore show that the ground state changes regularity, which depends qualitatively on the fractional power of the non-local operator.

math.SP↗

arXiv2022

Stability of ground state eigenvalues of non-local Schrödinger operators with respect to potentials and applications

In a first part of this paper we investigate the continuity (stability) of the spectrum of a class of non-local Schrödinger operators on varying the potentials. By imposing conditions of different strength on the convergence of the sequence of potentials, we give either direct proofs to show the strong or norm resolvent convergence of the so-obtained sequence of non-local Schrödinger operators, or via $Γ$-convergence of the related positive forms for more rough potentials. In a second part we use these results to show via a sequence of suitably constructed approximants that the ground states of massive or massless relativistic Schrödinger operators with spherical potential wells are radially decreasing functions.

math.AP↗

arXiv2022

Absence of Embedded Eigenvalues for Non-Local Schrödinger Operators

We consider non-local Schrödinger operators with kinetic terms given by several different types of functions of the Laplacian and potentials decaying to zero at infinity, and derive conditions ruling embedded eigenvalues out. Our goal in this paper is to advance techniques based on virial theorems, Mourre estimates, and an extended version of the Birman-Schwinger principle, previously developed for classical Schrödinger operators but thus far not used for non-local operators. We also present a number of specific cases by choosing particular classes of kinetic and potential terms, and discuss existence/non-existence of at-edge eigenvalues in a basic model case in function of the coupling parameter.

math.SP↗

arXiv2020

Hopf's lemma for viscosity solutions to a class of non-local equations with applications

We consider a large family of integro-differential equations and establish a non-local counterpart of Hopf's lemma, directly expressed in terms of the symbol of the operator. As closely related problems, we also obtain a variety of maximum principles for viscosity solutions. In our approach we combine direct analysis with functional integration, allowing a robust control around the boundary of the domain, and make use of the related ascending ladder height-processes. We then apply these results to a study of principal eigenvalue problems, the radial symmetry of the positive solutions, and the overdetermined non-local torsion equation.

math.AP↗

arXiv2020

Potentials for non-local Schrödinger operators with zero eigenvalues

The purpose of this paper is to give a systematic description of potentials decaying to zero at infinity, which generate eigenvalues at the edge of the absolutely continuous spectrum when combined with non-local operators defined by Bernstein functions of the Laplacian. By introducing suitable Hölder-Zygmund type spaces with different scale functions than usual, we study the action of these non-local Schrödinger operators in terms of second-order centered differences of eigenfunctions integrated with respect to singular kernels. First we obtain conditions under which the potentials decay at all, and are bounded continuous functions. Next we derive decay rates at infinity separately for operators with regularly varying and exponentially light Lèvy jump kernels. We show situations in which no decay occurs, implying that zero-energy eigenfunctions with specific decay properties cannot occur. Then we obtain detailed results on the sign of potentials at infinity which, apart from asymptotic behaviour at infinity, is a second main feature responsible for the occurrence or absence of zero eigenvalues. Finally, we study the behaviour of potentials at the origin, and analyze a delicate interplay between the pinning effect resulting from a well at zero combined with decay and sign at infinity, as a main mechanism in the formation of zero-energy bound states. Among the many possible examples of non-local operators, we single out the fractional Laplacian and the massive relativistic operator, and we will derive and make extensive use of an additive relationship between the two. In the paper we propose a unified framework and develop a purely analytic approach.

math.SP↗

arXiv2020

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 the 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.

math.AP↗

arXiv2019

Maximum principles and Aleksandrov-Bakelman-Pucci type estimates for non-local Schrödinger equations with exterior conditions

We consider Dirichlet exterior value problems related to a class of non-local Schrödinger operators, whose kinetic terms are given in terms of Bernstein functions of the Laplacian. We prove elliptic and parabolic Aleksandrov-Bakelman-Pucci type estimates, and as an application obtain existence and uniqueness of weak solutions. Next we prove a refined maximum principle in the sense of Berestycki-Nirenberg-Varadhan, and a converse. Also, we prove a weak anti-maximum principle in the sense of Clément-Peletier, valid on compact subsets of the domain, and a full anti-maximum principle by restricting to fractional Schrödinger operators. Furthermore, we show a maximum principle for narrow domains, and a refined elliptic ABP-type estimate. Finally, we obtain Liouville-type theorems for harmonic solutions and for a class of semi-linear equations. Our approach is probabilistic, making use of the properties of subordinate Brownian motion.

math.AP↗

arXiv2019

Universal Constraints on the Location of Extrema of Eigenfunctions of Non-Local Schrödinger Operators

We derive a lower bound on the location of global extrema of eigenfunctions for a large class of non-local Schrödinger operators in convex domains under Dirichlet exterior conditions, featuring the symbol of the kinetic term, the strength of the potential, and the corresponding eigenvalue, and involving a new universal constant. We show a number of probabilistic and spectral geometric implications, and derive a Faber-Krahn type inequality for non-local operators. Our study also extends to potentials with compact support, and we establish bounds on the location of extrema relative to the boundary edge of the support or level sets around minima of the potential.

math.SP↗

arXiv2018

Typical long time behaviour of ground state-transformed jump processes

We consider a class of Lévy-type processes derived via a Doob-transform from Lévy 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 Lévy generators perturbed by the potential. We derive precise lower and upper envelopes for the almost sure long time behaviour of these ground state-transformed Lévy processes, characterized through escape rates and integral tests. We also highlight the role of the parameters by specific examples.

math.PR↗

arXiv2018

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 of both $L^2$ and resonance solutions at infinity. 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.

math.FA↗

arXiv2018

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.

math.AP↗

arXiv2017

Kato's Inequality for Magnetic Relativistic Schrödinger Operators

Kato's inequality is shown for the magnetic relativistic Schrödinger operator $H_{A,m}$ defined as the operator theoretical {\it square root} of the selfadjoint, magnetic nonrelativistic Schrödinger operator $(-i\nabla-A(x))^2+m^2$ with an $L^{2}_{\text{\rm loc}}$ vector potential $A(x)$.

math.AP↗

arXiv2016

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 Lévy intensity. Our approach is based on path integration and probabilistic potential theory techniques, and all results are also illustrated by specific examples.

math.PR↗

arXiv2016

Transition in the decay rates of stationary distributions of Lévy 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.

cond-mat.stat-mech↗

arXiv2015

Contractivity and ground state domination properties for non-local Schrödinger operators

We study supercontractivity and hypercontractivity of Markov semigroups obtained via ground state transformation of non-local Schrödinger operators based on generators of symmetric jump-paring Lévy processes with Kato-class confining potentials. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps, and the related operators include pseudo-differential operators of interest in mathematical physics. We refine these contractivity properties by the concept of $L^p$-ground state domination and its asymptotic version, and derive sharp necessary and sufficient conditions for their validity in terms of the behaviour of the Lévy density and the potential at infinity. As a consequence, we obtain for a large subclass of confining potentials that, on the one hand, supercontractivity and ultracontractivity, on the other hand, hypercontractivity and asymptotic ultracontractivity of the transformed semigroup are equivalent properties. This is in stark contrast to classical Schrödinger operators, for which all these properties are known to be different.

math.PR↗

arXiv2015

Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman-Kac semigroups for a class of Lévy processes

We introduce a class of Lévy 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 $λ$-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 Lévy 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.

math.PR↗

arXiv2013

The Spectrum of Non-Local Discrete Schroedinger Operators with a delta-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 $C^1$-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.

math-ph↗