Integrability of the one dimensional Schrödinger equation

Abstract

We present a definition of integrability for the one dimensional Schroedinger equation, which encompasses all known integrable systems, i.e. systems for which the spectrum can be explicitly computed. For this, we introduce the class of rigid functions, built as Liouvillian functions, but containing all solutions of rigid differential operators in the sense of Katz, and a notion of natural of boundary conditions. We then make a complete classification of rational integrable potentials. Many new integrable cases are found, some of them physically interesting. (1) is the quantum equivalent of a one degree of freedom Hamiltonian system. In classical mechanics, this system is always integrable in the sense that we are always able to express the solutions in terms of quadrature. In this quantum equivalent, it is no longer the case.There has been various ways to define the meaning of V quantum integrable. Here we focus on complete integrability, i.e. finding all the eigenfunctions, which is more restrictive to partial integrability cases, as in [1] where only finitely many eigenstates are found. The closest notion to the one presented in this article is the following Definition 1 (Integrability definition in [2]). The equation (1) is said to be integrable if for all E in the spectrum, the solutions of equation (1) are Liouvillian.This definition seems to contain all "quantum integrable" cases, at least in the case of discrete spectrum. However, there are several inconvenience • This does not allow (at least a priori) to compute the set S. Indeed, the integrability check has to be done the other way. Assume we know the spectrum S, the system is integrable if and only if equation (1) has Liouvillian solutions. This can however, for a fixed E, be done algorithmically through the Kovacic algorithm [3].• The notion of integrability is strongly dependant of the boundary conditions. This happens for example in the following equationIf the boundary condition is being square integrable on R + , then the system is integrable with a discrete spectrum S. If we look for square integrable solutions on R − there are none, and if only near 0 then S = C. In this continuous case, the system is not integrable in the above sense.

Dynamics and non-integrability of the double spring pendulum

“…The Morales-Ramis theory has already been successfully applied to various important physical systems [53][54][55][56][57][58][59][60] as well as non-Hamiltonian systems [56,[61][62][63], to cite just a few. Because of this, many new integrable and super-integrable systems were found [64][65][66].…”

“…The classification of the subgroups of Sp(4, C) is known. Among other things, this fact was used in [57] where the equivalent of the Kovacic algorithm for symplectic differential operators of degree four was formulated.…”

Functional Renormalization analysis of Bose-Einstien Condensation through complex interaction in Harmonic Oscillator; Can Bendixson criteria be extended to complex time?

“…The integrability of the generic quantum problem in 1 dimension strongly depend on the boundary conditions at some points and potential form which is detailed in the article 33 ,Where a theorem has proved for a integrable quantum potential say γ comes from a generic function…”

“…) in general except z all are constants which are related to the theorem 1 and 2 of the article 33 later we get these solutions for the wave functions from various methods and we do have the following proposition to get the limit cycle in wave function real-imaginary plane.Note that the Hypergeometric functions are related together [34][35][36] and in special limits can be expressed as product of incomplete gamma and gamma functions and they are analytic(expanded as series).We do have different corrections to these forms in later section but we don't claim these are general to any complex form of the potential.For one of the case we get M(z, c) = − 1 z + az + 1 F 1 1 F 1 2z 1−b which can be shown for all of the cases these are indeed a bethe ansatz form…”

Liouvillian solutions for second order linear differential equations with polynomial coefficients

“…In a more general sense, the study of second order linear differential equations with polynomial coefficients can be done through Kovacic's algorithm, see [11] and see also [10], as well by Asymptotic Iteration Method., see [7]. Recently Combot in [9] developed another method to obtain exactly solvable potentials, in the sense of Natanzon, involving rigid functions in the sense of Katz.…”