95 research outputs found
Differential Galois Theory of Linear Difference Equations
We present a Galois theory of difference equations designed to measure the
differential dependencies among solutions of linear difference equations. With
this we are able to reprove Hoelder's Theorem that the Gamma function satisfies
no polynomial differential equation and are able to give general results that
imply, for example, that no differential relationship holds among solutions of
certain classes of q-hypergeometric functions.Comment: 50 page
Galoisian Approach to integrability of Schr\"odinger Equation
In this paper, we examine the non-relativistic stationary Schr\"odinger
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
Schr\"odinger 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.Comment: 62 page
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