Potential theory, path integrals and the Laplacian of the indicator

@article{Lange2012PotentialTP,
  title={Potential theory, path integrals and the Laplacian of the indicator},
  author={Rutger-Jan Lange},
  journal={Journal of High Energy Physics},
  year={2012},
  volume={2012},
  pages={1-46}
}
  • Rutger-Jan Lange
  • Published 2012
  • Physics, Mathematics
  • Journal of High Energy Physics
A bstractThis 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/σ2) and ħ = 1. The Laplacian of the indicator can be interpreted using the theory of distributions: it is the d-dimensional analogue of the… Expand

Figures from this paper

Point and surface interactions in quantum mechanics: resolving the paradox
This paper develops a distributional theory for the Schrödinger equation with point interactions in d = 1, and surface interactions in d > 1. Currently, there is no generally accepted method wherebyExpand
An explicit realization of resonant-tunnelling δ″-potentials
A family of non-trivial one-point interactions is constructed by means of a two-scale approximation with spatially symmetric piecewise constant functions. The approximation is realized through twoExpand
Contact Interactions in One-Dimensional Quantum Mechanics: a Family of Generalized б'-Potentials
A “one-point” approximation is proposed to investigate the transmission of electrons through the extra thin heterostructures composed of two parallel plane layers. The typical example is the bilayerExpand
Point Interactions With Bias Potentials
We develop an approach on how to define single-point interactions under the application of external fields. The essential feature relies on an asymptotic method based on the one-point approximationExpand
The topological molecule: Its finite fluxes, exchange stability and minimal surfaces
Molecules have at least one nontrivial topological property in common: their minimal surfaces of finite flux. This is why they are stable aggregates of atoms mutually engaged to varying degrees viaExpand
Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials
Several families of one-point interactions are derived from the system consisting of two and three δ-potentials which are regularized by piecewise constant functions. In physical terms such anExpand
Geometric perturbation theory and Acoustic Boundary Condition Dynamics
TLDR
The relaxation dynamics of the tympanic-membrane system, which neuronal information processing stems from, is explicitly obtained in first order and both the initial and the quasi-stationary asymptotic state are derived and analyzed. Expand
Adaptation and validation of FFT methods for homogenization of lattice based materials
An FFT framework which preserves a good numerical performance in the case of domains with large regions of empty space is proposed and analyzed for its application to lattice based materials. TwoExpand
Elastic models of dislocations based on atomistic Kanzaki forces
© 2018 authors. Published by the American Physical Society. This paper studies the relationship between the atomistic representation of crystalline dislocations as Kanzaki forces and the continuumExpand
Smoothed particle hydrodynamics continuous boundary force method for Navier-Stokes equations subject to a Robin boundary condition
TLDR
Considering the no-slip boundary condition as a special case of the slip boundary condition, it is demonstrated that the SPH-CBF method accurately describes both theNo-Slip and slip conditions. Expand
...
1
2
3
...

References

SHOWING 1-10 OF 61 REFERENCES
Solution of the Schrodinger Equation in Terms of Classical Paths
Abstract An expression in terms of classical paths is derived for the Laplace transform Ω(s) of the Green function G of the Schrodinger equation with respect to 1 h . For an analytic potential V(r),Expand
Boundary dynamics and multiple reflection expansion for Robin boundary conditions
In the presence of a boundary interaction, Neumann boundary conditions should be modified to contain a function S of the boundary fields: $({\ensuremath{\nabla}}_{N}+S)\ensuremath{\varphi}=0.$Expand
Path integrals for potential problems with δ-function perturbation
The author presents several examples of potential problems with a δ-function perturbation by means of path integrals. The idea is to sum a perturbation series expansion resulting in anExpand
Two . dimensional Brownian Motion and Harmonic Functions
    1. The purpose of this paper is to investigate the properties of two-dimensional Brownian motions’ and to apply the results thus obtained to the theory of harmonic functions in the Gaussian plane.Expand
    The heat equation and reflected Brownian motion in time-dependent domains
    The paper is concerned with reflecting Brownian motion (RBM) in domains with deterministic moving boundaries, also known as "noncylindrical domains,'' and its connections with partial differentialExpand
    Quantum Field Theory
    Quantum field theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the standard model, are formulated. Quantum electrodynamics (QED),Expand
    Distribution of eigenfrequencies for the wave equation in a finite domain: III. Eigenfrequency density oscillations
    Abstract This paper is concerned with the oscillations which appear in the smoothed density of eigenvalues when the smoothing width is relatively small. The existence of these oscillations isExpand
    Techniques and Applications of Path Integration
    Partial table of contents: Probabilities and Probability Amplitudes for Paths. Correspondence Limit for the Path Integral (Heuristic). Vector Potentials and Another Proof of the Path IntegralExpand
    A Laplace operator with boundary conditions singular at one point
    We discuss a recent paper of Berry and Dennis (J. Phys. A: Math. Theor. 2008 41 135203) concerning a Laplace operator on a smooth domain with singular boundary condition. We explain a paradox in theExpand
    THE PATH DECOMPOSITION EXPANSION AND MULTIDIMENSIONAL TUNNELING
    Abstract This paper consists of two main topics. 1. (i) The path decomposition expansion: a new path integral technique which allows us to break configuration space into disjoint regions and expressExpand
    ...
    1
    2
    3
    4
    5
    ...