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arXiv · 1011.1642

Spectral/quadrature duality: Picard-Vessiot theory and finite-gap potentials

Yurii V. Brezhnev
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Abstract

In the framework of differential Galois theory we treat the classical spectral problem $\Psi"-u(x)\Psi=\lambda\Psi$ and its finite-gap potentials as exactly solvable in quadratures by Picard--Vessiot without involving special functions; the ideology goes back to the 1919 works by J. Drach. We show that duality between spectral and quadrature approaches is realized through the Weierstrass permutation theorem for a logarithmic Abelian integral. From this standpoint we inspect known facts and obtain new ones: an important formula for the $\Psi$-function and $\Theta$-function extensions of Picard--Vessiot fields. In particular, extensions by Jacobi's $\theta$-functions lead to the (quadrature) algebraically integrable equations for the $\theta$-functions themselves.

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Yurii V. Brezhnev. 2010-11-07. Spectral/quadrature duality: Picard-Vessiot theory and finite-gap potentials. https://doi.org/10.1090/conm%2F563%2F11162

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