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

Bulletin of the South Ural State University Series Mathematical Modelling Programming and Computer Software

On a Limit Pass from Two-Point to One-Point Interaction in a One Dimensional Quantum Mechanical Problem Giving Rise to a Spontaneous Symmetry Breaking

Universidad de Antofagasta, A. Restuccia, Universidad Simon Bolvar, A. Sotomayor, Universidad de Antofagasta, Vladimir Strauss, Universidad Simon Bolvar

Published Jan 1, 2021• 0 citations
Publisher

Abstract

We analyze, by means of singular potentials defined in terms of Dirac functions and their derivatives, a one dimensional symmetry breaking in quantum mechanics.From a mathematical point of view we use a technique of selfadjoint extensions applied to a symmetric differential operator with a domain containing smooth functions which vanish at two inner points of the real line.As is well known, the latter leads to a two-point boundary problem.We compute the resolvent of the corresponding extension and investigate its behavior in the case in which the inner points change their positions.The domain of these extensions can contain some functions with non differentiability or discontinuity at the points mentioned before.This fact can be interpreted as a presence of singular potentials like shifted Dirac delta functions and/or their first derivative centered at the same points.Then, we study the existence of broken-symmetry bound states.For some given entanglement boundary conditions we can show the existence of a ground state, which leads to a spontaneous symmetry breaking.We also prove that within a frame of Pontryagin spaces this type of symmetry breaking is saved if the distance between the mentioned above interior points tends to zero and then we can reformulate this result in terms of a larger Hilbert space.

DOI
10.14529/mmp210106

Related papers

On a Model of Spontaneous Symmetry Breaking in Quantum Mechanics

Universidad de Antofagasta, A. Restuccia, Universidad Simon Bolivar +4 more

Our goal is to find a model for the phenomenon of spontaneous symmetry breaking arising in one dimensional quantum mechanical problems.For this purpose we consider boundary value problems related with two interior points of the real line, symmetric with respect to the origin.This approach can be treated as a presence of singular potentials containing shifted Dirac delta functions and their derivatives.From mathematical point of view we use a technique of selfadjoint extensions applied to a symmetric differential operator which has a domain containing smooth functions vanishing in two mentioned above points.We calculate the resolvent of corresponding extension and investigate its behavior if the interior points change their positions.The domain of these extensions can contain some functions that have non differentiability or discontinuity at the points mentioned above, the latter can be interpreted as an appearance of singular potentials centered at the same points.Next, broken-symmetry bound states are discovered.More precisely, for a particular entanglement of boundary conditions, there is a ground state, generating a spontaneous symmetry breaking, stable under the phenomenon of decoherence provoked from external fluctuations.We discuss the model in the context of the "chiral" broken-symmetry states of molecules like N H 3 .We show that within a Hilbert space approach a spontaneous symmetry breaking disappears if the distance between the mentioned above interior points tends to zero.

Bulletin of the South Ural State University Series Mathematical Modelling Programming and Computer Software•Jan 1, 2020•1 citation

Entanglement of boundary conditions giving rise to spontaneous symmetry breaking in quantum mechanics

A. Restuccia, A. Sotomayor, Vladimir Strauss

We study spontaneous symmetry breaking in one dimensional quantum mechanical problems in terms of two-point boundary problems which lead to singular potentials containing Dirac delta functions and its derivatives. We search for broken-symmetry bound states. For a particular entanglement of boundary conditions, we show the existence of a ground state, giving rise to a spontaneous symmetry breaking, stable under the phenomenon of decoherence generated from external fluctuations. We discuss the model in the context of the "chiral" broken-symmetry states of molecules like $NH_3$.

arXiv:1902.05836•Feb 15, 2019•0 citations

Non-local interactions in quantum mechanics modelled by shifted Dirac delta functions

A. Restuccia, A. Sotomayor, Vladimir Strauss

We introduce a new interaction formulated in terms of two shifted delta functions with non-local boundary conditions. For this purpose we use a method of self-adjoint extensions for a particular simmetric operator. We obtain bound states of the associated quantum mechanical problem and study its symmetry properties. We compare the results with the ground state of the quantum mechanical problem with a local interaction in terms of two delta functions.

Journal of Physics Conference Series•Jun 1, 2018•2 citations

Point and surface interactions in quantum mechanics

Rutger‐Jan Lange

Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrodinger's equation. This paper, in contrast, investigates the integral form of Schrodinger's equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schrodinger's integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schrodinger's differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov's result to hypersurfaces. Second, we derive a new closed-form solution to Schrodinger's integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schrodinger's differential equation. Third, we derive boundary conditions for `super-singular' potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution, and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schrodinger's integral equation is viable tool for studying singular interactions in quantum mechanics.

•Jan 29, 2014•0 citations

Point interactions: boundary conditions or potentials with the Dirac delta function

Salvatore De Vincenzo, Carlet Sánchez

We study the problem of a nonrelativistic quantum particle moving on a real line with an idealized and localized singular interaction with zero range at x = 0 (i.e., a point interaction there). This kind of system can be described in two ways: (i) by considering an alternative free system (i.e., without the singular potential) but excluding the point x = 0 (In this case, the point interaction is exclusively encoded in the boundary conditions.) and (ii) by explicitly considering the singular interaction by means of a local singular potential. In this paper we relate, compare, and discuss, in a simple and pedagogical way these two equivalent approaches. Our main goal in this paper is to introduce the essential ideas about point interactions in a very accesible form to advanced undergraduates.

Canadian Journal of Physics•Nov 1, 2010•9 citations

Self-Adjointness and the Renormalization of Singular Potentials

Sarang Gopalakrishnan, William Loinaz

Schrodinger operators with very singular potentials fail to produce reasonable spectra, because the usual boundary conditions are insufficient to make them self-adjoint. If one regularizes a singular potential at a length , the bound state energies diverge as → 0. A meaningful spectrum can be restored by picking a self-adjoint extension of the operator, by renormalizing the theory, or by allowing nonunitary time evolution. We show that renormalizations of the 1/r potential fall into two classes: those that are asymptotically equivalent to self-adjoint extensions, and those that reduce to treatments of delta functions. We also apply the apparatus of self-adjoint extensions and renormalization to clarify aspects of anomalous symmetry breaking, supersymmetric quantum mechanics, and one-dimensional quantum mechanics. Lines composed upon reading a thesis abstract by S. Gopalakrishnan Think a moment, dear reader, of that long-legged fly, The sprung meniscus underfoot, And what asymptotes invisible to well-trained eyes Still break the surface at its root. How a singular potential picks tight-bounded energy Apart till it no longer recognizes Its own mirrored face, its extensions, or symmetry; Unless one cautiously renormalizes. You have tried in these pensive pages to show the stuff Such theory’s made of, its quiddity When logic is applied to the diamond rough, And quantum notes ring purely. May Apollo’s clarity pierce through your reasoning And self-adjointed truth and fact Elevate your thought to that sense so pleasing When truth recovers nature’s pact.

•Jan 1, 2006•12 citations
We analyze, by means of singular potentials defined in terms of Dirac functions and their derivatives, a one dimensional symmetry breaking in quantum mechanics.From a mathematical point of view we use a technique of selfadjoint extensions applied to a symmetric differential operator with a domain containing smooth functions which vanish at two inner points of the real line.As is well known, the latter leads to a two-point boundary problem.We compute the resolvent of the corresponding extension and investigate its behavior in the case in which the inner points change their positions.The domain of these extensions can contain some functions with non differentiability or discontinuity at the points mentioned before.This fact can be interpreted as a presence of singular potentials like shifted Dirac delta functions and/or their first derivative centered at the same points.Then, we study the existence of broken-symmetry bound states.For some given entanglement boundary conditions we can show the existence of a ground state, which leads to a spontaneous symmetry breaking.We also prove that within a frame of Pontryagin spaces this type of symmetry breaking is saved if the distance between the mentioned above interior points tends to zero and then we can reformulate this result in terms of a larger Hilbert space.
Our goal is to find a model for the phenomenon of spontaneous symmetry breaking arising in one dimensional quantum mechanical problems.For this purpose we consider boundary value problems related with two interior points of the real line, symmetric with respect to the origin.This approach can be treated as a presence of singular potentials containing shifted Dirac delta functions and their derivatives.From mathematical point of view we use a technique of selfadjoint extensions applied to a symmetric differential operator which has a domain containing smooth functions vanishing in two mentioned above points.We calculate the resolvent of corresponding extension and investigate its behavior if the interior points change their positions.The domain of these extensions can contain some functions that have non differentiability or discontinuity at the points mentioned above, the latter can be interpreted as an appearance of singular potentials centered at the same points.Next, broken-symmetry bound states are discovered.More precisely, for a particular entanglement of boundary conditions, there is a ground state, generating a spontaneous symmetry breaking, stable under the phenomenon of decoherence provoked from external fluctuations.We discuss the model in the context of the "chiral" broken-symmetry states of molecules like N H 3 .We show that within a Hilbert space approach a spontaneous symmetry breaking disappears if the distance between the mentioned above interior points tends to zero.
We study spontaneous symmetry breaking in one dimensional quantum mechanical problems in terms of two-point boundary problems which lead to singular potentials containing Dirac delta functions and its derivatives. We search for broken-symmetry bound states. For a particular entanglement of boundary conditions, we show the existence of a ground state, giving rise to a spontaneous symmetry breaking, stable under the phenomenon of decoherence generated from external fluctuations. We discuss the model in the context of the "chiral" broken-symmetry states of molecules like NH3NH_3NH3​.
Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrodinger's equation. This paper, in contrast, investigates the integral form of Schrodinger's equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schrodinger's integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schrodinger's differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov's result to hypersurfaces. Second, we derive a new closed-form solution to Schrodinger's integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schrodinger's differential equation. Third, we derive boundary conditions for `super-singular' potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution, and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schrodinger's integral equation is viable tool for studying singular interactions in quantum mechanics.
We study the problem of a nonrelativistic quantum particle moving on a real line with an idealized and localized singular interaction with zero range at x = 0 (i.e., a point interaction there). This kind of system can be described in two ways: (i) by considering an alternative free system (i.e., without the singular potential) but excluding the point x = 0 (In this case, the point interaction is exclusively encoded in the boundary conditions.) and (ii) by explicitly considering the singular interaction by means of a local singular potential. In this paper we relate, compare, and discuss, in a simple and pedagogical way these two equivalent approaches. Our main goal in this paper is to introduce the essential ideas about point interactions in a very accesible form to advanced undergraduates.
Schrodinger operators with very singular potentials fail to produce reasonable spectra, because the usual boundary conditions are insufficient to make them self-adjoint. If one regularizes a singular potential at a length , the bound state energies diverge as → 0. A meaningful spectrum can be restored by picking a self-adjoint extension of the operator, by renormalizing the theory, or by allowing nonunitary time evolution. We show that renormalizations of the 1/r potential fall into two classes: those that are asymptotically equivalent to self-adjoint extensions, and those that reduce to treatments of delta functions. We also apply the apparatus of self-adjoint extensions and renormalization to clarify aspects of anomalous symmetry breaking, supersymmetric quantum mechanics, and one-dimensional quantum mechanics. Lines composed upon reading a thesis abstract by S. Gopalakrishnan Think a moment, dear reader, of that long-legged fly, The sprung meniscus underfoot, And what asymptotes invisible to well-trained eyes Still break the surface at its root. How a singular potential picks tight-bounded energy Apart till it no longer recognizes Its own mirrored face, its extensions, or symmetry; Unless one cautiously renormalizes. You have tried in these pensive pages to show the stuff Such theory’s made of, its quiddity When logic is applied to the diamond rough, And quantum notes ring purely. May Apollo’s clarity pierce through your reasoning And self-adjointed truth and fact Elevate your thought to that sense so pleasing When truth recovers nature’s pact.