Galoisian approach to integrability of Schrödinger equation

Abstract

Abstract. In this paper, we examine the non-relativistic stationary Schrödinger equation from a differential Galois-theoretic perspective. The main algorithmic tools are pullbacks of second order ordinary linear differential operators, so as to achieve rational function coefficients ("algebrization"), and Kovacic's algorithm for solving the resulting equations. In particular, we use this Galoisian approach to analyze Darboux transformations, Crum iterations and supersymmetric quantum mechanics. We obtain the ground states, eigenvalues, eigenfunctions, eigenstates and differential Galois groups of a large class of Schrödinger equations, e.g. those with exactly solvable and shape invariant potentials (the terms are defined within). Finally, we introduce a method for determining when exact solvability is possible.

Nonintegrability of the Armbruster--Guckenheimer--Kim Quartic Hamiltonian Through Morales--Ramis Theory

“…(16) and Eq. (17) are no-integrables because they correspond to quantic harmonic oscillators with zero energy level for both planes Γ 1 and Γ 3 , see [3,6]. Finally, the cases h =ĥ = 0 are not considered because we obtain for both planes Γ 1 and Γ 3 that the normal variational equations (16) and (17) are trivially integrables.…”

“…Finally, the cases h =ĥ = 0 are not considered because we obtain for both planes Γ 1 and Γ 3 that the normal variational equations (16) and (17) are trivially integrables. The first one corresponds to a differential equation with constant coefficients and the second one corresponds to a Cauchy-Euler equation, which are always integrable with abelian differential Galois group, see [4,6]. Thus Morales-Ramis Theorem does not give any obstruction to the integrability.…”

Differential Galois Groups and Representation of Quivers for Seismic Models With Constant Hessian of Square of Slowness

“…One of these applications corresponds to Morales-Ramis theory (see [32]), that is, the differential Galois theory linked with the non-integrability of dynamical systems, being the starting point to prove non-integrability of Hamiltonian and non-Hamiltonian systems, see [1][2][3][4][5]11]. Similarly, differential Galois theory has been used to study integrability and nonintegrability of polynomial vector fields (see [7,14]), integrability in quantum mechanics (see [6][7][8][9][10]12]) and integrability in quantum optics (see [13]). …”

On integrable rational potentials of the Dirac equation

“…in [2,[4][5][6]. When it comes to the Dirac equation, the complete analysis in the case of general polynomial potentials was carried out in a previous work by one of the authors [7].…”