- Nonlocal Operators

2013 ◽  
pp. 292-321
Keyword(s):  
Nonlocal Operators

scholarly journals Local versus nonlocal elliptic equations: short-long range field interactions

2020 ◽  
Vol 10 (1) ◽  
pp. 895-921
Author(s):  
Daniele Cassani ◽  
Luca Vilasi ◽  
Youjun Wang
Keyword(s):  
Asymptotic Behavior ◽  
Maximum Principle ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Spectral Properties ◽  
Fractional Laplacian ◽  
Coupling Parameter ◽  
Nonlocal Operators ◽  
The Maximum Principle ◽  
Regularity Results

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

scholarly journals A path integral formulation for particle detectors: the Unruh-DeWitt model as a line defect

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Ivan M. Burbano ◽  
T. Rick Perche ◽  
Bruno de S. L. Torres
Keyword(s):  
Path Integral ◽  
Degrees Of Freedom ◽  
Wilson Line ◽  
Transition Probability ◽  
Relativistic Quantum ◽  
Quantum Fields ◽  
Line Defect ◽  
Particle Detectors ◽  
Nonlocal Operators ◽  
Detector Model

Abstract Particle detectors are an ubiquitous tool for probing quantum fields in the context of relativistic quantum information (RQI). We formulate the Unruh-DeWitt (UDW) particle detector model in terms of the path integral formalism. The formulation is able to recover the results of the model in general globally hyperbolic spacetimes and for arbitrary detector trajectories. Integrating out the detector’s degrees of freedom yields a line defect that allows one to express the transition probability in terms of Feynman diagrams. Inspired by the light-matter interaction, we propose a gauge invariant detector model whose associated line defect is related to the derivative of a Wilson line. This is another instance where nonlocal operators in gauge theories can be interpreted as physical probes for quantum fields.

scholarly journals Regularity and rigidity theorems for a class of anisotropic nonlocal operators

2016 ◽  
Vol 153 (1-2) ◽  
pp. 53-70 ◽  
Author(s):  
Alberto Farina ◽  
Enrico Valdinoci
Keyword(s):  
Nonlocal Operators ◽  
Rigidity Theorems

scholarly journals Non-radial functions, nonlocal operators and Markov processes over p-adic numbers

2019 ◽  
Vol 24 (2) ◽  
pp. 381-406 ◽  
Author(s):  
Leonardo Fabio Chacón-Cortés ◽  
Oscar Francisco Casas-Sánchez
Keyword(s):  
Markov Processes ◽  
First Passage Time ◽  
Parabolic Type ◽  
Passage Time ◽  
Fundamental Solutions ◽  
Nonlocal Operators ◽  
Transition Functions ◽  
Time Problem ◽  
The Cauchy Problem ◽  
First Passage Time Problem

The main goal of this article is to study a new class of nonlocal operators and the Cauchy problem for certain parabolic-type pseudodifferential equations naturally associated with them. The fundamental solutions of these equations are transition functions of Markov processes on an n-dimensional vector space over the p-adic numbers. We also study some properties of these Markov processes, including the first passage time problem.

scholarly journals Regularity estimates for elliptic nonlocal operators

2020 ◽  
Vol 13 (2) ◽  
pp. 317-370 ◽  
Author(s):  
Bartłomiej Dyda ◽  
Moritz Kassmann
Keyword(s):  
Nonlocal Operators ◽  
Regularity Estimates
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