# Spectral-quadrature duality: Picard-Vessiot theory and finite-gap potentials - Mathematics > Classical Analysis and ODEs

Abstract: In the framework of differential Galois theory we treat the classicalspectral problem $\Psi-ux\Psi=\lambda\Psi$ and its finite-gap potentials asexactly solvable in quadratures by Picard-Vessiot without involving specialfunctions; the ideology goes back to the 1919 works by J. Drach. We show thatduality between spectral and quadrature approaches is realized through theWeierstrass permutation theorem for a logarithmic Abelian integral. From thisstandpoint we inspect known facts and obtain new ones: an important formula forthe $\Psi$-function and $\Theta$-function extensions of Picard-Vessiot fields.In particular, extensions by Jacobi-s $\theta$-functions lead to thequadrature algebraically integrable equations for the $\theta$-functionsthemselves.

Author: Yurii V. Brezhnev

Source: https://arxiv.org/