Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Letters

Boundary Conditions from Path Integrals

T. Jaroszewicz

Published Nov 21, 1988• 3 citations
Publisher

Abstract

A particle moving in two dimensions in a repulsive ${r}^{\ensuremath{-}2}$ potential, and in the presence of a magnetic flux line, is examined as an example of a system for which the Schr\"odinger Hamiltonian is not essentially self-adjoint. The path-integral propagator is, nevertheless, shown to exist, and to define a unique self-adjoint extension of the Hamiltonian, corresponding to specific boundary conditions at the origin. Physical implications are discussed.

DOI
10.1103/physrevlett.61.2401

Related papers

The functional integral for a free particle on a half-plane

Michel Carreau

A free nonrelativistic particle moving in two dimensions on a half-plane can be described by self-adjoint Hamiltonians characterized by boundary conditions imposed on the systems. The most general boundary condition is parameterized in terms of the elements of an infinite-dimensional matrix. We construct the Brownian functional integral for each of these self-adjoint Hamiltonians. Nonlocal boundary conditions are implemented by allowing the paths striking the boundary to jump to other locations on the boundary. Analytic continuation in time results in the Green’s functions of the Schrödinger equation satisfying the boundary condition characterizing the self-adjoint Hamiltonian.

arXiv:hep-th/9208052•Journal of Mathematical Physics•Dec 1, 1992•13 citations

Self-adjoint extensions of the Hamiltonian for a charged particle in the presence of a thread of magnetic flux

Francisco Antônio Bezerra Coutinho, Yukihisa Nogami, J. Fernando Perez

Self-adjoint extensions of the usual Hamiltonian are studied for a charged particle moving in the x-y plane in the presence of an infinitely thin thread of magnetic flux along the z axis. This is done by starting with a square-well potential and taking its zero-range limit. The wave function for such an extended Hamiltonian is, although square integrable, singular at the origin. There are interesting differences between the cases with and without the magnetic flux.

Physical Review A•Nov 1, 1992•21 citations

The functional integral for a free particle on a half-plane

Michel Carreau

(D.O.E.) under contract #DE-AC02-76ER03069, and by the Fonds pour la Formation de Chercheurs et l’Aide à la Recherche. A free non-relativistic particle moving in two dimensions on a half-plane can be described by self-adjoint Hamiltonians characterized by boundary conditions imposed on the systems. The most general boundary condition is parameterized in terms of the elements of an infinite-dimensional matrix. We construct the Brownian functional integral for each of these self-adjoint Hamiltonians. Non-local boundary conditions are implemented by allowing the paths striking the boundary to jump to other locations on the boundary. Analytic continuation in time results in the Green’s functions of the Schrödinger equation satisfying the boundary condition characterizing the self-adjoint Hamiltonian. 1 I.

•Jan 1, 1992•9 citations

Self-Adjoint Extension Approach for Singular Hamiltonians in (2 + 1) Dimensions

Vinícius Salem, Ramon F. Costa, Edilberto Oliveira Silva +1 more

In this work, we review two methods used to approach singular Hamiltonians in (2+1) dimensions. Both methods are based on the self-adjoint extension approach. It is very common to find singular Hamiltonians in quantum mechanics, especially in quantum systems in the presence of topological defects, which are usually modelled by point interactions. In general, it is possible to apply some kind of regularization procedure, as the vanishing of the wave function at the location of the singularity, ensuring that the wave function is square integrable and then can be associated with a physical state. However, an analysis based on the self-adjoint extension approach can lead to more general boundary conditions that still gives acceptable physical states. We exemplify the methods by exploring the bound and scattering scenarios of a spin 1/2 charged particle with an anomalous magnetic moment in the Aharonov-Bohm potential in the conical space.

arXiv:1910.07105•Frontiers in Physics•Nov 6, 2019•6 citations

Magnetic perturbations of anyonic and Aharonov–Bohm Schrödinger operators

Michele Correggi, Davide Fermi

We study the Hamiltonian describing two anyons moving in a plane in the presence of an external magnetic field and identify a one-parameter family of self-adjoint realizations of the corresponding Schrödinger operator. We also discuss the associated model describing a quantum particle immersed in a magnetic field with a local Aharonov–Bohm singularity. For a special class of magnetic potentials, we provide a complete classification of all possible self-adjoint extensions.

arXiv:2006.09056•Journal of Mathematical Physics•Mar 1, 2021•16 citations

On the most general boundary conditions for the Aharonov-Bohm scattering of a Dirac particle: helicity and Aharonov-Bohm symmetry conservation

Vanilse S. Araujo, Francisco Antônio Bezerra Coutinho, J. Fernando Perez

We show that the Hamiltonian H and the helicity operator Λ of a Dirac particle moving in two dimensions in the presence of an infinitely thin magnetic flux tube each admit a four-parameter family of self-adjoint extensions. Each extension is in one-to-one correspondence with the boundary conditions (BCs) to be satisfied by the eigenfunctions at the origin. Although the actions of these two operators commute before specification of BCs, to ensure helicity conservation it is not sufficient to take the same BCs for both operators. We show that, given certain relations between the parameters of the extensions, it is possible to write down the most general domain where both operators H and Λ are self-adjoint with helicity conservation and also Aharonov-Bohm symmetry (ϕ→ϕ + 1) preserved, where ϕ is the magnetic flux in natural units. The continuity of the dynamics is also obtained. Our results imply that neither helicity conservation nor Aharonov-Bohm symmetry by itself solves the problem of choosing the `physical' BCs for this system.

Journal of Physics A Mathematical and General•Oct 15, 2001•11 citations
A particle moving in two dimensions in a repulsive r\ensuremath−2{r}^{\ensuremath{-}2}r\ensuremath−2 potential, and in the presence of a magnetic flux line, is examined as an example of a system for which the Schr"odinger Hamiltonian is not essentially self-adjoint. The path-integral propagator is, nevertheless, shown to exist, and to define a unique self-adjoint extension of the Hamiltonian, corresponding to specific boundary conditions at the origin. Physical implications are discussed.
A free nonrelativistic particle moving in two dimensions on a half-plane can be described by self-adjoint Hamiltonians characterized by boundary conditions imposed on the systems. The most general boundary condition is parameterized in terms of the elements of an infinite-dimensional matrix. We construct the Brownian functional integral for each of these self-adjoint Hamiltonians. Nonlocal boundary conditions are implemented by allowing the paths striking the boundary to jump to other locations on the boundary. Analytic continuation in time results in the Green’s functions of the Schrödinger equation satisfying the boundary condition characterizing the self-adjoint Hamiltonian.
(D.O.E.) under contract #DE-AC02-76ER03069, and by the Fonds pour la Formation de Chercheurs et l’Aide à la Recherche. A free non-relativistic particle moving in two dimensions on a half-plane can be described by self-adjoint Hamiltonians characterized by boundary conditions imposed on the systems. The most general boundary condition is parameterized in terms of the elements of an infinite-dimensional matrix. We construct the Brownian functional integral for each of these self-adjoint Hamiltonians. Non-local boundary conditions are implemented by allowing the paths striking the boundary to jump to other locations on the boundary. Analytic continuation in time results in the Green’s functions of the Schrödinger equation satisfying the boundary condition characterizing the self-adjoint Hamiltonian. 1 I.
In this work, we review two methods used to approach singular Hamiltonians in (2+1) dimensions. Both methods are based on the self-adjoint extension approach. It is very common to find singular Hamiltonians in quantum mechanics, especially in quantum systems in the presence of topological defects, which are usually modelled by point interactions. In general, it is possible to apply some kind of regularization procedure, as the vanishing of the wave function at the location of the singularity, ensuring that the wave function is square integrable and then can be associated with a physical state. However, an analysis based on the self-adjoint extension approach can lead to more general boundary conditions that still gives acceptable physical states. We exemplify the methods by exploring the bound and scattering scenarios of a spin 1/2 charged particle with an anomalous magnetic moment in the Aharonov-Bohm potential in the conical space.
We show that the Hamiltonian H and the helicity operator Λ of a Dirac particle moving in two dimensions in the presence of an infinitely thin magnetic flux tube each admit a four-parameter family of self-adjoint extensions. Each extension is in one-to-one correspondence with the boundary conditions (BCs) to be satisfied by the eigenfunctions at the origin. Although the actions of these two operators commute before specification of BCs, to ensure helicity conservation it is not sufficient to take the same BCs for both operators. We show that, given certain relations between the parameters of the extensions, it is possible to write down the most general domain where both operators H and Λ are self-adjoint with helicity conservation and also Aharonov-Bohm symmetry (ϕ→ϕ + 1) preserved, where ϕ is the magnetic flux in natural units. The continuity of the dynamics is also obtained. Our results imply that neither helicity conservation nor Aharonov-Bohm symmetry by itself solves the problem of choosing the `physical' BCs for this system.