Discrete rotational symmetry in quantum systems distributed on a ring arises when the Hamiltonian remains invariant under rotations of 2π/N, where N is the number of equally spaced sites, particles, or potentials. This discrete structure defines the cyclic group CN, whose representation in the state space determines fundamental properties of the energy spectrum and the system's dynamics. In quantum mechanics, such symmetry implies that the eigenvalues of the rotation operator are phases of the type ei2πm/N, where m is the quantum number associated with the discrete angular momentum.Discrete rotational symmetry in quantum systems distributed on a ring arises when the Hamiltonian remains invariant under rotations of 2π/N, where N is the number of equally spaced sites, particles, or potentials. This discrete structure defines the cyclic group CN, whose representation in the state space determines fundamental properties of the energy spectrum and the system's dynamics. In quantum mechanics, such symmetry implies that the eigenvalues of the rotation operator are phases of the type ei2πm/N, where m is the quantum number associated with the discrete angular momentum.

Apr 9, 20260 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.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:1712.10209Dec 29, 20173 citations

Discrete spectrum of many-particle quantum systems not having stable subsystems. II

Semjon A. Vugalter, G. M. Zhislin

Functional Analysis and Its ApplicationsJul 1, 19800 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/S1061920825600850The 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 PhysicsSep 1, 20252 citations

\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