Quasi-exact solvability and entropies of the one-dimensional regularised Calogero model

Abstract

The divergence in the interaction term of the Calogero model can be prevented introducing a cutoff length parameter, this modification leads to a quasi-exactly solvable model whose eigenfunctions can be written in terms of Heun's polynomials. It is shown both, analytical and numerically.that the reduced density matrix obtained tracing out one particle from the two-particle density operator can be obtained exactly as well as its entanglement spectrum. The number of non-zero eigenvalues in these cases is finite. Besides, it is shown that taking the limit in which the cutoff distance goes to zero, the reduced density matrix and finite entanglement spectrum of the Calogero model is retrieved. The entanglement Rényi entropy is also studied to characterize the physical traits of the model. It is found that the quasi-exactly solvable character of the model is put into evidence by the entanglement entropies when they are calculated numerically over the parameter space of the model.

Convexity properties of superpositions of degenerate bipartite eigenstates

“…As has been said in the Introduction, the study of the Calogero model was what triggered the formulation of the criterion that is the object of the present work. Fortunately, there is a growing number of quasi-exactly solvable models [33] that can be used to study properties of strongly interacting two-particle models [34].…”

Two electrons in harmonic confinement coupled to light in a cavity

“…For hydrogen-like models, solutions have been found for particular forms of the inhomogeneous magnetic fields 26,27 . Examples of other known QES models include the planar Dirac electron in hydrogen-like atoms 28,29 , one-body problems in power-law central potentials 30,31 , relativistic 2D pion in constant magnetic fields 32 , and 1D and 3D regularized Calogero models 33,34 . QES models with different forms of confinements, e.g.…”