# 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} }

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…

## 31 Citations

Point and surface interactions in quantum mechanics: resolving the paradox

- Mathematics
- 2014

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 whereby…

An explicit realization of resonant-tunnelling δ″-potentials

- Mathematics
- 2015

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 two…

2D Schrödinger operators with singular potentials concentrated near curves

- Mathematics, Computer ScienceApplicable Analysis
- 2020

The transmission conditions on $\gamma$ for the eigenfunctions of the Schr\"{o}dinger operators reveal a nontrivial connection between spectral properties of $H_\varepsilon$ and the geometry of $\Gamma$.

Distribution theory for Schrödinger’s integral equation

- Mathematics
- 2015

Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrödinger’s equation. This paper, in contrast, investigates the integral form of…

Contact Interactions in One-Dimensional Quantum Mechanics: a Family of Generalized б'-Potentials

- PhysicsUkrainian Journal of Physics
- 2019

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 bilayer…

Point Interactions With Bias Potentials

- PhysicsFront. Phys.
- 2019

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 approximation…

The topological molecule: Its finite fluxes, exchange stability and minimal surfaces

- Physics
- 2016

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 via…

Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials

- Physics
- 2017

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 an…

Geometric perturbation theory and Acoustic Boundary Condition Dynamics

- PhysicsPhysica D: Nonlinear Phenomena
- 2020

Adaptation and validation of FFT methods for homogenization of lattice based materials

- Computer ScienceComputer Methods in Applied Mechanics and Engineering
- 2022

## References

SHOWING 1-10 OF 61 REFERENCES

Boundary dynamics and multiple reflection expansion for Robin boundary conditions

- Mathematics
- 2002

In the presence of a boundary interaction, Neumann boundary conditions should be modified to contain a function S of the boundary fields: $({\ensuremath{\nabla}}_{N}+S)\ensuremath{\varphi}=0.$…

Two . dimensional Brownian Motion and Harmonic Functions

- Mathematics

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.…

The heat equation and reflected Brownian motion in time-dependent domains

- Mathematics
- 2004

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 differential…

Quantum Field Theory

- Physics
- 1999

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),…

Distribution of eigenfrequencies for the wave equation in a finite domain: III. Eigenfrequency density oscillations

- Mathematics
- 1972

Techniques and Applications of Path Integration

- Physics
- 1981

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 Integral…

A Laplace operator with boundary conditions singular at one point

- Mathematics
- 2009

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 the…

On hearing the shape of a drum: further results

- Mathematics
- 1971

1. Introduction. The underlying problem is to deduce the shape of a drum or plane uniform membrane from the knowledge of its spectrum of eigenvalues ωn = i√λn. It has been shown by Kac(3) that some…