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

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.

Major changes (23pp -> 32pp); Final version; LaTeX

References in corpus (2)

• On a Schwarzian PDE associated with the KdV Hierarchy
• On the Dubrovin Equations for the Finite-gap Potentials

Cited by in corpus (3)

• Nonlinear supersymmetry in the quantum Calogero model
• Rational KdV Potentials and Differential Galois Theory
• An exactly solvable problem of wave fronts and applications to the asymptotic theory