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

Russian Journal of Mathematical Physics

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

M. A. Lyalinov

Published Sep 1, 2025• 2 citations
Publisher

Abstract

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

DOI
10.1134/s1061920825600850

Related papers

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

Three one-dimensional quantum particles scattering problem with short-range repulsive pair potentials. To the question of absolutely continuous spectrum eigenfunctions asymptotics justification

Alexander Mikhajlovich Budylin, Сергей Борисович Левин

We consider the quantum scattering problem of three one-dimensional particles with repulsive shortrange pair potentials. For clarity, we restrict ourself by the case of finite pair potentials. The absence of singular continuous spectrum of the corresponding Schröedinger operator for a broad class of pair potentials was proved earlier in a known work of E. Mourre. Nevertheless, the Mourre techniques do not allow description of the asymptotics of absolutely continuous spectrum eigenfunctions. In this work, regardless of Mourre results, we prove the existence of the resolvent limit values on absolutely continuous spectrum and construct them explicitly. It allows us, following the known procedure, to derive the absolutely continuous spectrum eigen-functions asymptotics, suggested earlier in works of V. S. Buslaev and his co-authors. Our approach, close to the foundational work of L. D. Faddeev devoted to three-dimensional particles, specifically uses the ideas of Schwarz alternating method.

Days on Diffraction•Jun 1, 2016•0 citations

Point interaction Hamiltonians in bounded domains

Blanchard, Philippe, Rodolfo Figari, Andrea Mantile

Making use of recent techniques in the theory of self-adjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a three-dimensional bounded domain with regular boundaries. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator together with the corresponding eigenfunctions. These operators are used to build up a model system where the dynamics of a quantum particle depends on the state of a quantum bit.

•Jan 1, 2007•13 citations

Point interaction Hamiltonians in bounded domains

Ph. Blanchard, R. Figari, Andrea Mantile

Making use of recent techniques in the theory of self-adjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a three-dimensional bounded domain with regular boundaries. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator together with the corresponding eigenfunctions. These operators are used to build up a model system where the dynamics of a quantum particle depends on the state of a quantum bit.

arXiv:0704.3249•Journal of Mathematical Physics•Aug 1, 2007•10 citations

Spectral Properties of the 2+1 Fermionic Trimer with Contact Interactions

Simon Becker, Alessandro Michelangeli, Andrea Ottolini

We qualify the main features of the spectrum of the Hamiltonian of point interaction for a three-dimensional quantum system consisting of three point-like particles, two identical fermions, plus a third particle of different species, with two-body interaction of zero range. For arbitrary magnitude of the interaction, and arbitrary value of the mass parameter (the ratio between the mass of the third particle and that of each fermion) above the stability threshold, we identify the essential spectrum, localise and prove the finiteness of the discrete spectrum, qualify the angular symmetry of the eigenfunctions, and prove the monotonicity of the eigenvalues with respect to the mass parameter. We also demonstrate the existence of bound states in a physically relevant regime of masses.

arXiv:1712.10209•Dec 29, 2017•3 citations

New approach to numerical computation of the eigenfunctions of the continuous spectrum of three-particle Schrödinger operator: I. One-dimensional particles, short-range pair potentials

V. S. Buslaev, S. B. Levin, Pekka Neittaanmäki +1 more

Basing on analogy between the three-body scattering problem and the diffraction problem of the plane wave (for the case of the short range pair potentials) by the system of six half transparent screens, we presented a new approach to the few-body scattering problem. The numerical results have been obtained for the case of the short range nonnegative pair potentials. The presented method allows a natural generalization to the case of the long range pair potentials.

arXiv:0909.4529•Journal of Physics A Mathematical and Theoretical•Jun 16, 2010•14 citations
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\deltaδ -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 consider the quantum scattering problem of three one-dimensional particles with repulsive shortrange pair potentials. For clarity, we restrict ourself by the case of finite pair potentials. The absence of singular continuous spectrum of the corresponding Schröedinger operator for a broad class of pair potentials was proved earlier in a known work of E. Mourre. Nevertheless, the Mourre techniques do not allow description of the asymptotics of absolutely continuous spectrum eigenfunctions. In this work, regardless of Mourre results, we prove the existence of the resolvent limit values on absolutely continuous spectrum and construct them explicitly. It allows us, following the known procedure, to derive the absolutely continuous spectrum eigen-functions asymptotics, suggested earlier in works of V. S. Buslaev and his co-authors. Our approach, close to the foundational work of L. D. Faddeev devoted to three-dimensional particles, specifically uses the ideas of Schwarz alternating method.
Making use of recent techniques in the theory of self-adjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a three-dimensional bounded domain with regular boundaries. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator together with the corresponding eigenfunctions. These operators are used to build up a model system where the dynamics of a quantum particle depends on the state of a quantum bit.
Making use of recent techniques in the theory of self-adjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a three-dimensional bounded domain with regular boundaries. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator together with the corresponding eigenfunctions. These operators are used to build up a model system where the dynamics of a quantum particle depends on the state of a quantum bit.
We qualify the main features of the spectrum of the Hamiltonian of point interaction for a three-dimensional quantum system consisting of three point-like particles, two identical fermions, plus a third particle of different species, with two-body interaction of zero range. For arbitrary magnitude of the interaction, and arbitrary value of the mass parameter (the ratio between the mass of the third particle and that of each fermion) above the stability threshold, we identify the essential spectrum, localise and prove the finiteness of the discrete spectrum, qualify the angular symmetry of the eigenfunctions, and prove the monotonicity of the eigenvalues with respect to the mass parameter. We also demonstrate the existence of bound states in a physically relevant regime of masses.