Advances in Differential Equations

Transformation theory of symmetric differential expressions

Horst Behncke and Don Hinton

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We consider the problem of transforming a symmetric differential expression of even or odd order with both real and complex coefficients with a Kummer-Liouville transformation. An existence proof is given which yields an algorithm for computing exactly the coefficients of the transformed equation. By using the concept of a modified Kummer-Liouville transformation we derive explicit expressions for the coefficients of the transformed equation which are correct modulo Levinson terms only. However, the application of Levinson's theorem to asymptotic integration shows that Levinson terms have no effect on the asymptotics of the eigenfunctions.

Article information

Adv. Differential Equations, Volume 11, Number 6 (2006), 601-626.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34A30: Linear equations and systems, general
Secondary: 34C20: Transformation and reduction of equations and systems, normal forms 34L05: General spectral theory 34L20: Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions 47E05: Ordinary differential operators [See also 34Bxx, 34Lxx] (should also be assigned at least one other classification number in section 47)


Behncke, Horst; Hinton, Don. Transformation theory of symmetric differential expressions. Adv. Differential Equations 11 (2006), no. 6, 601--626.

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