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

arXiv:math-ph/9910029

Some properties of the one-dimensional generalized point interactions (a torso)

Pavel Exner, Harald Grosse

Published Oct 19, 1999• 28 citations
arXivHTMLPDF

Abstract

This text is a part of an unfinished project which deals with the generalized point interaction (GPI) in one dimension. We employ two natural parametrizations, which are known but have not attracted much attention, to express the resolvent of the GPI Hamiltonian as well as its spectral and scattering properties. It is also shown that the GPI yields one of the simplest models in which a non-trivial Berry phase is exhibited. Furthermore, the generalized Kronig-Penney model corresponding to the GPI is discussed. We show that there are three different types of the high-energy behaviour for the corresponding band spectrum.

DOI
10.48550/arxiv.math-ph/9910029

Related papers

Scattering from generalized point interactions using self-adjoint extensions in Pontryagin spaces

J. F. van Diejen, A. Tip

Generalized point interactions (GPI) appearing formally in (quantum) physics as pseudopotentials are given an operator theoretic basis. It is pointed out that in contrast to ordinary point interactions these GPI models are not self-adjoint generators in the physical Hilbert space ℋ0, but ‘‘live’’ instead in an apparently nonphysical Pontryagin space of the type Πm=ℋ0⊕Cm■Cm. The physical interpretation of the models lies in the association of a unitary S-operator on ℋ0 within a two-space setting. A general formalism for GPI is presented and used to answer questions, which were raised in the literature, on the underlying mathematical structure of some models involving pseudopotentials in higher dimensions. Quantum mechanical scattering from GPI, with the purpose of replacing a spherically symmetric short-range interaction potential in the low-energy regime, is discussed in detail. The possible power of using GPI to enable explicit calculations in quantum mechanical and electromagnetic multiple scattering from distortions that are small relative to the wavelength is briefly mentioned.

Journal of Mathematical Physics•Mar 1, 1991•47 citations

Three Identical One-Dimensional Quantum Particles with Point Interaction as a Solvable Model: III. Positive Essential Spectrum

M. A. Lyalinov

Abstract This is the third part of our work dealing with spectral analysis for the Hamiltonian of three identical one-dimensional quantum particles. It deals with positive energies and the corresponding generalized eigenfunctions. The asymptotic behavior of the generalized eigenfunctions is studied and interpreted in terms of the diffraction theory.

Russian Journal of Mathematical Physics•Jun 1, 2026•0 citations

Generalized One-Dimensional Point Interaction in Relativistic and Non-relativistic Quantum Mechanics

Takaomi Shigehara, Hiroshi Mizoguchi, Mishima, T. +1 more

We first give the solution for the local approximation of a four parameter family of generalized one-dimensional point interactions within the framework of non-relativistic model with three neighboring $δ$ functions. We also discuss the problem within relativistic (Dirac) framework and give the solution for a three parameter family. It gives a physical interpretation for so-called $ε$ potential. It will be also shown that the scattering properties at high energy substantially differ between non-relativistic and relativistic cases.

arXiv:quant-ph/9911059•Nov 13, 1999•0 citations

Three Identical One-Dimensional Quantum Particles with Point Interaction as a Solvable Model: I. Discrete Spectrum

M. A. Lyalinov

The paper deals with a Hamiltonian, namely, with a semi-bounded self-adjoint operator that is attributed to the problem of scattering of three one-dimensional particles with point interaction in pairs, in other words, with $$ \delta $$ -functional singular potential of interaction. The support of the potential in the Hamiltonian coincides with a symmetric star-graph having six leads on the two-dimensional plane. Due to the symmetry, we find that such a model is exactly solvable, which means that the eigenfunctions of the discrete spectrum and the generalized eigenfunctions of the essential (absolutely continuous) spectrum are determined explicitly, i.e., by quadrature. In this (first part) of our work we describe the discrete spectrum and the eigenfunctions. DOI 10.1134/S1061920825600850

Russian Journal of Mathematical Physics•Sep 1, 2025•2 citations

A class of explicitly soluble, local, many-center Hamiltonians for one-particle quantum mechanics in two and three dimensions. I

Alex Grossmann, Raphael J. Høegh-Krohn, M. Mebkhout

We derive an explicit formula for the resolvent of a class of one-particle, many-center, local Hamiltonians. This formula gives, in particular, a full description of a model molecule given by point interactions at n arbitrarily placed fixed centers in three dimensions. It also gives a three−dimensional analog of the Kronig–Penney model.

Journal of Mathematical Physics•Sep 1, 1980•83 citations

Zak phase and bulk-boundary correspondence in a generalized Dirac-Kronig-Penney model

Giuliano Angelone, Domenico Monaco, Gabriele Peluso

We investigate the topological properties of a generalized Dirac--Kronig--Penney model, a continuum one-dimensional model for a relativistic quantum chain. By tuning the coupling parameters this model can accommodate five Altland--Zirnbauer--Cartan symmetry classes, three of which (AIII, BDI and D) support non-trivial topological phases in dimension one. We characterize analytically the spectral properties of the Hamiltonian in terms of a spectral function, and numerically compute the Zak phase to probe the bulk topological content of the insulating phases. Our findings reveal that, while the Zak phase is quantized in classes AIII and BDI, it exhibits non-quantized values in class D, challenging its traditional role as a topological marker in continuum settings. We also discuss the bulk-boundary correspondence for a truncated version of the chain, analyzing how the emergence of edge states depends on both the truncation position and the boundary conditions. In classes AIII and BDI, we find that the Zak phase effectively detects edge states as a relative boundary topological index, although the correspondence is highly sensitive to the parameters characterizing the truncation.

quant-pharXiv:2602.03378•Feb 4, 2026
This text is a part of an unfinished project which deals with the generalized point interaction (GPI) in one dimension. We employ two natural parametrizations, which are known but have not attracted much attention, to express the resolvent of the GPI Hamiltonian as well as its spectral and scattering properties. It is also shown that the GPI yields one of the simplest models in which a non-trivial Berry phase is exhibited. Furthermore, the generalized Kronig-Penney model corresponding to the GPI is discussed. We show that there are three different types of the high-energy behaviour for the corresponding band spectrum.
Generalized point interactions (GPI) appearing formally in (quantum) physics as pseudopotentials are given an operator theoretic basis. It is pointed out that in contrast to ordinary point interactions these GPI models are not self-adjoint generators in the physical Hilbert space ℋ0, but ‘‘live’’ instead in an apparently nonphysical Pontryagin space of the type Πm=ℋ0⊕Cm■Cm. The physical interpretation of the models lies in the association of a unitary S-operator on ℋ0 within a two-space setting. A general formalism for GPI is presented and used to answer questions, which were raised in the literature, on the underlying mathematical structure of some models involving pseudopotentials in higher dimensions. Quantum mechanical scattering from GPI, with the purpose of replacing a spherically symmetric short-range interaction potential in the low-energy regime, is discussed in detail. The possible power of using GPI to enable explicit calculations in quantum mechanical and electromagnetic multiple scattering from distortions that are small relative to the wavelength is briefly mentioned.
We first give the solution for the local approximation of a four parameter family of generalized one-dimensional point interactions within the framework of non-relativistic model with three neighboring δδδ functions. We also discuss the problem within relativistic (Dirac) framework and give the solution for a three parameter family. It gives a physical interpretation for so-called εεε potential. It will be also shown that the scattering properties at high energy substantially differ between non-relativistic and relativistic cases.
The paper deals with a Hamiltonian, namely, with a semi-bounded self-adjoint operator that is attributed to the problem of scattering of three one-dimensional particles with point interaction in pairs, in other words, with δ \delta δ -functional singular potential of interaction. The support of the potential in the Hamiltonian coincides with a symmetric star-graph having six leads on the two-dimensional plane. Due to the symmetry, we find that such a model is exactly solvable, which means that the eigenfunctions of the discrete spectrum and the generalized eigenfunctions of the essential (absolutely continuous) spectrum are determined explicitly, i.e., by quadrature. In this (first part) of our work we describe the discrete spectrum and the eigenfunctions. DOI 10.1134/S1061920825600850
We investigate the topological properties of a generalized Dirac--Kronig--Penney model, a continuum one-dimensional model for a relativistic quantum chain. By tuning the coupling parameters this model can accommodate five Altland--Zirnbauer--Cartan symmetry classes, three of which (AIII, BDI and D) support non-trivial topological phases in dimension one. We characterize analytically the spectral properties of the Hamiltonian in terms of a spectral function, and numerically compute the Zak phase to probe the bulk topological content of the insulating phases. Our findings reveal that, while the Zak phase is quantized in classes AIII and BDI, it exhibits non-quantized values in class D, challenging its traditional role as a topological marker in continuum settings. We also discuss the bulk-boundary correspondence for a truncated version of the chain, analyzing how the emergence of edge states depends on both the truncation position and the boundary conditions. In classes AIII and BDI, we find that the Zak phase effectively detects edge states as a relative boundary topological index, although the correspondence is highly sensitive to the parameters characterizing the truncation.