Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory

Abstract

In this paper we start with proving that the Schrödinger equation (SE) with the classical 12 − 6 Lennard-Jones (L-J) potential is nonintegrable in the sense of the differential Galois theory (DGT), for any value of energy; i.e., there are no solutions in closed form for such differential equation. We study the 10−6 potential through DGT and SUSYQM; being it one of the two partner potentials built with a superpotential of the form w(r) ∝ 1/r 5 . We also find that it is integrable in the sense of DGT for zero energy. A first analysis of the applicability and physical consequences of the model is carried out in terms of the so called De Boer principle of corresponding states. A comparison of the second virial coefficient B(T ) for both potentials shows a good agreement for low temperatures. As a consequence of these results we propose the 10 − 6 potential as an integrable alternative to be applied in further studies instead of the original 12 − 6 L-J potential. Finally we study through DGT and SUSYQM the integrability of the SE with a generalized (2ν − 2) − ν L-J potential. This analysis do not include the study of square integrable wave functions, excited states and energies different than zero for the generalization of L-J potentials.

From Poincaré to May: The Genesis of Discrete Dynamics

“…Dynamical systems are very important in the development of different areas, see for example [1] for applications in Quantum Mechanics and see also [5,6,7,8] for applications in Classical Mechanics. In particular, discrete dynamical systems play an important role due to they are an useful tool to understand continuous systems by decreasing its complexity, either by diminish dimensions or passing from continuous to discrete time.…”

Liouvillian solutions for second order linear differential equations with polynomial coefficients

“…In the first case if m is even then (r, m) ∈ C 2 ∪ C 3 , otherwise due to Kovacic algorithm to be an exclusive procedure (r, m) necessarily has to be in C 2 . Finally, the last case never occur because step two of Kovacic algorithm requires the quantity d = 1 2 (−m − 2q − 1) to be a non-negative integer which clearly is not, so we have to discard the family of equations with coefficients in M (2q+1,m) .…”

“…In section 4 present some applications that involve the analysis of biconfluent Heun equation and doubly confluent Heun equation, as well Schrödinger Equations with Mie potentials (Laurent polynomial potentials, exponential potentials) and Inverse Square Root potentials, see [1,8,9]. Finally, our results allow us to state that there are no new algebraically solvable Laurent polynomial potentials for the Shcrödinger equation beyond those previously known (Corolary 4.1) corresponding to m = 2 and r = 0, 1, 2.…”

“…The space M (r,m) of monic Laurent polynomials of type (r, m) is an affine algebraic variety M (r,m) ≃ C * × C r+m−1 . The purpose of this paper is to classify Picard-Vessiot integrable equations in the family (1) and to study how their Liouvillian solutions depend algebraically on the coefficients of L(x) when it moves in the space M (r,m) . We thus introduce the spectral set.…”