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On a family of Hill's equations in the complex field

โœ Scribed by Amar Makhlouf; Fernand Nahon

Publisher
Springer Netherlands
Year
1991
Tongue
English
Weight
472 KB
Volume
52
Category
Article
ISSN
1572-9478

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โœฆ Synopsis

The variational equation of the periodic solution in an homogenous potential of degree 2m leads to a family of Hill's equations. H. Yoshida solves these equations by a transformation to the Gauss equation of hypergeometric functions. We study directly this family for complex t, and in the case m -----2, we give a detailed description of critical solutions.

R4sum6. L'6quation variationeUe de la solution p6riodique dans un potentiel homogene de degr6 2rn conduit ?~ une famiUe d'6quations de Hill. H. Yoshida r6soud ces 4quations en les transformant en 6quation de Gauss des fonctions hyperg6om6triques. Nous les 6tudions directement pour t complexe, et dans le cas m = 2 nous donnons une description d6taill6e des solutions critiques.

๐Ÿ“œ SIMILAR VOLUMES

On the eigenvalues of Hill's equation
On the eigenvalues of Hill's equation
โœ L. E. Blumenson ๐Ÿ“‚ Article ๐Ÿ“… 1963 ๐Ÿ› John Wiley and Sons ๐ŸŒ English โš– 253 KB
On the Floquet Exponents of Hill's Equat
On the Floquet Exponents of Hill's Equation Systems
โœ Robert Denk ๐Ÿ“‚ Article ๐Ÿ“… 1995 ๐Ÿ› John Wiley and Sons ๐ŸŒ English โš– 292 KB ๐Ÿ‘ 1 views

For the k x k-matrix-valued version of Hill's equation it is shown that the dimension of the matrix needed to compute the Floquet exponents can be reduced from 2k to k. Also the existence of periodic solutions is equivalent to the non-invertibility of certain k x k-matrices.