On the negative point spectrum of quantum mechanical three-body systems

Jose Iorio

Rodolfo Figari, Hamidreza Saberbaghi, Alessandro Teta

Abstract We re-investigate the entire family of many center point interaction Hamiltonians. Under the assumption of exchange symmetry with respect to the point positions, we show that a large sub-family of point interaction Hamiltonian operators does not become either singular or trivial when the positions of two or more scattering centers tend to coincide. In this sense, they appear to be renormalized by default as opposed to the point interaction Hamiltonians usually considered in the literature. Functions in their domains satisfy regularized boundary conditions which turn out to be very similar to the ones proposed recently in many-body quantum mechanics to define three-particle system Hamiltonians with contact interactions bounded from below. In the two-center case, we study the behavior of the negative eigenvalues as a function of the center distance. The result is used to analyze a formal Born–Oppenheimer approximation of a three-particle system with two heavy bosons and one light particle. We demonstrate that this simplified model describes a stable system (no ‘fall to the center’ problem is present). Furthermore, in the unitary limit, the energy spectrum is characterized by an infinite sequence of negative energy eigenvalues accumulating at zero according to the geometrical Efimov law.

G. Karl, Edward Obryk

Dinkar C. Khandekar, S. V. Lawande

A one-dimensional problem of three equal particles interacting pairwise via a potential that is the sum of a quadratic and an inversely quadratic term is discussed from the point of view of classical mechanics. It is shown that the quantum energy spectrum can be obtained from the classical theory of action-angle variables and Bohr-Sommerfeld rules.

V. S. Olkhovsky

Jan Michael Rost, D. Wintgen

A highly accurate calculation is supplemented by an adiabatic approximation to explore the resonance spectrum of positronium ${\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}$. Surprisingly, the spectrum can be understood and classified with ${\mathrm{H}}_{2}^{+}$ quantum numbers by treating the interelectronic axis of ${\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}$ as an adiabatic parameter. We report and interpret the existence of $^{1}$S shape resonances, a phenomenon so far unknown in three-body Coulomb systems. The new results on ${\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}$ combined with previous results for ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ suggest the existence of a resonance spectrum and its similarity for all ABA Coulomb systems with charges \ensuremath{\Vert}${\mathit{Z}}_{\mathit{A}}$/${\mathit{Z}}_{\mathit{B}}$\ensuremath{\Vert}=1 and masses ${\mathit{m}}_{\mathit{A}}$/${\mathit{m}}_{\mathit{B}}$\ensuremath{\ge}1.