Methods and Applications of Analysis

Volume 19 (2012)

Number 3

Galoisian approach for a Sturm-Liouville problem on the infinite interval

Pages: 267 – 288



David Blázquez-Sanz (Escuela de Matemáticas, Universidad Sergio Arboleda, Bogotá, Colombia)

Kazuyuki Yagasaki (Mathematics Division, Department of Information Engineering, Niigata University, Japan)


We study a Sturm-Liouville type eigenvalue problem for second-order differential equations on the infinite interval (−∞, ∞). Here the eigenfunctions are nonzero solutions exponentially decaying at infinity. We prove that at any discrete eigenvalue the differential equations are integrable in the setting of differential Galois theory under general assumptions. Our result is illustrated with three examples for a stationary Schrödinger equation having a generalized Hulthén potential; a linear stability equation for a traveling front in the Allen-Cahn equation; and an eigenvalue problem related to the Lamé equation.


Sturm-Liouville problem, differential Galois theory, integrability, monodromy group

2010 Mathematics Subject Classification

34B09, 34B24, 35B35, 81Q05

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