The Experts below are selected from a list of 33 Experts worldwide ranked by ideXlab platform
Rutgerjan Lange - One of the best experts on this subject based on the ideXlab platform.
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potential theory path Integrals and the laplacian of the indicator
arXiv: Mathematical Physics, 2013Co-Authors: Rutgerjan LangeAbstract:This paper links the field of potential theory -- i.e. the Dirichlet and Neumann problems for the heat and Laplace equation -- to that of the Feynman path Integral, by postulating that the potential is equal to plus/minus the Laplacian of the indicator of the domain D. The Laplacian of the indicator is a generalized function: it is the d-dimensional analogue of the Dirac delta'-function. This function has -- according to the author's best knowledge -- not formally been defined before. We show, first, that the path Integral's perturbation series (or Born series) matches the classical single and double boundary layer series of potential theory, thereby connecting two hitherto unrelated fields. Second, we show that the perturbation series is valid for all domains D that allow Green's theorem (i.e. with a finite number of corners, edges and cusps), thereby expanding the classical applicability of boundary layers. Third, we show that the minus (plus) in the potential holds for the Dirichlet (Neumann) boundary condition; showing for the first time a particularly close connection between these two classical problems. Fourth, we demonstrate that the perturbation series of the path Integral Converges in a monotone/alternating fashion, depending on the convexity/concavity of the domain. We also discuss the third boundary problem (which poses Robin boundary conditions) and discuss an extension to moving domains.
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potential theory path Integrals and the laplacian of the indicator
Journal of High Energy Physics, 2012Co-Authors: Rutgerjan LangeAbstract:This paper links the field of potential theory — i.e. the Dirichlet and Neumann problems for the heat and Laplace equation — to that of the Feynman path Integral, by postulating the following seemingly ill-defined potential: $ V(x):=\mp \frac{{{\sigma^2}}}{2}\nabla_x^2{1_{{x\in D}}} $ where the volatility is the reciprocal of the mass (i.e. m = 1/σ 2) and ħ = 1. The Laplacian of the indicator can be interpreted using the theory of distributions: it is the d-dimensional analogue of the Dirac δ′-function, which can formally be defined as $ \partial_x^2{1_{x>0 }} $ . We show, first, that the path Integral's perturbation series (or Born series) matches the classical single and double boundary layer series of potential theory, thereby connecting two hitherto unrelated fields. Second, we show that the perturbation series is valid for all domains D that allow Green's theorem (i.e. with a finite number of corners, edges and cusps), thereby expanding the classical applicability of boundary layers. Third, we show that the minus (plus) in the potential holds for the Dirichlet (Neumann) boundary condition; showing for the first time a particularly close connection between these two classical problems. Fourth, we demonstrate that the perturbation series of the path Integral Converges as follows: We also discuss the third boundary problem (which poses Robin boundary conditions) and discuss an extension to moving domains.
A Vasudeva S Murthy - One of the best experts on this subject based on the ideXlab platform.
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phase space feynman path Integrals of higher order parabolic type with general functional as integrand
Bulletin Des Sciences Mathematiques, 2015Co-Authors: Naoto Kumanogo, A Vasudeva S MurthyAbstract:Abstract We give a general class of functionals for which the phase space Feynman path Integrals of higher order parabolic type have a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of the phase space path Integral Converges uniformly on compact subsets with respect to the endpoint of position paths and to the starting point of momentum paths. Our class of functionals is rich because it is closed under addition and multiplication. The interchange of the order with the integration with respect to time, the interchange of the order with a limit and the perturbation expansion formula hold in the path Integrals.
Fujiwara Daisuke - One of the best experts on this subject based on the ideXlab platform.
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Rigorous time slicing approach to Feynman path Integrals
'Springer Science and Business Media LLC', 2017Co-Authors: Fujiwara DaisukeAbstract:This book proves that Feynman's original definition of the path Integral actually Converges to the fundamental solution of the Schrödinger equation at least in the short term if the potential is differentiable sufficiently many times and its derivatives of order equal to or higher than two are bounded. The semi-classical asymptotic formula up to the second term of the fundamental solution is also proved by a method different from that of Birkhoff. A bound of the remainder term is also proved. The Feynman path Integral is a method of quantization using the Lagrangian function, whereas Schrödinger's quantization uses the Hamiltonian function. These two methods are believed to be equivalent. But equivalence is not fully proved mathematically, because, compared with Schrödinger's method, there is still much to be done concerning rigorous mathematical treatment of Feynman's method. Feynman himself defined a path Integral as the limit of a sequence of Integrals over finite-dimensional spaces which is obtained by dividing the time interval into small pieces. This method is called the time slicing approximation method or the time slicing method. This book consists of two parts. Part I is the main part. The time slicing method is performed step by step in detail in Part I. The time interval is divided into small pieces. Corresponding to each division a finite-dimensional Integral is constructed following Feynman's famous paper. This finite-dimensional Integral is not absolutely convergent. Owing to the assumption of the potential, it is an oscillatory Integral. The oscillatory Integral techniques developed in the theory of partial differential equations are applied to it. It turns out that the finite-dimensional Integral gives a finite definite value. The stationary phase method is applied to it. Basic properties of oscillatory Integrals and the stationary phase method are explained in the book in detail. Those finite-dimensional Integrals form a sequence of approximation of the Feynman path Integral when the division goes finer and finer. A careful discussion is required to prove the convergence of the approximate sequence as the length of each of the small subintervals tends to 0. For that purpose the book uses the stationary phase method of oscillatory Integrals over a space of large dimension, of which the detailed proof is given in Part II of the book. By virtue of this method, the approximate sequence Converges to the limit. This proves that the Feynman path Integral Converges. It turns out that the convergence occurs in a very strong topology. The fact that the limit is the fundamental solution of the Schrödinger equation is proved also by the stationary phase method. The semi-classical asymptotic formula naturally follows from the above discussion. A prerequisite for readers of this book is standard knowledge of functional analysis. Mathematical techniques required here are explained and proved from scratch in Part II, which occupies a large part of the book, because they are considerably different from techniques usually used in treating the Schrödinger equation
Burglind Joricke - One of the best experts on this subject based on the ideXlab platform.
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the logarithmic Integral Converges
1994Co-Authors: Victor Havin, Burglind JorickeAbstract:Divergence of a logarithmic Integral is the principal sufficient condition of many forms of the UP. But in fact it coincides with the necessary condition. The logarithmic Integral determines a border-line separating two realms, the one undividedly governed by the UP (and described in Chapter 2) and the other where the resistance to the UP is possible and non-zero pairs (f, f) are allowed whose elements are small simultaneously. This chapter is devoted to methods of construction (or at least to the existence proofs) of such pairs.
Naoto Kumanogo - One of the best experts on this subject based on the ideXlab platform.
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phase space feynman path Integrals of higher order parabolic type with general functional as integrand
Bulletin Des Sciences Mathematiques, 2015Co-Authors: Naoto Kumanogo, A Vasudeva S MurthyAbstract:Abstract We give a general class of functionals for which the phase space Feynman path Integrals of higher order parabolic type have a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of the phase space path Integral Converges uniformly on compact subsets with respect to the endpoint of position paths and to the starting point of momentum paths. Our class of functionals is rich because it is closed under addition and multiplication. The interchange of the order with the integration with respect to time, the interchange of the order with a limit and the perturbation expansion formula hold in the path Integrals.