On integrability of finite-gap potentials

Details On integrability of finite-gap potentials On integrability of finite-gap potentials

2005-05-05

arxiv.org/abs/nlin/0505015v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

28 pages; In Russian. Major conceptual changes + new results. 6 theorems and 4 propositions were added. Much improved. English

Scientific paper

For spectral problems, determined by ordinary differential equations, we consider finite-gap potentials as exact solvable by quadratures in the spirit of the Picard--Vessio theory and suggest that this class is the only one. Ideology goes back to works by Drach 1919 but has not been mentioned in the modern literature. We exhibit new examples and technique for obtaining necessary ingredients of exact integrability: new formula for the $\Psi$-function, main trace formula and problem of inversion of Abelian integrals.

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