Spectral/quadrature duality: Picard-Vessiot theory and finite-gap potentials. (English) Zbl 1248.81032

Brezhnev, Yurii V.

Acosta-Humánez, Primitivo B. (ed.) et al., Algebraic aspects of Darboux transformations, quantum integrable systems and supersymmetric quantum mechanics, Jairo Charris Seminar 2010, Santa Maria, Colombia, August 2010. Providence, RI: American Mathematical Society (AMS); Bogota: Instituto de Matemáticas y sus Aplicaciones (IMA). (ISBN 978-0-8218-7584-1/pbk). Contemporary Mathematics 563, 1-31 (2012).

Summary: 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 [C. R. 168, 47–50 (1919; JFM 47.0411.03); C. R. 168, 337–340 (1919; JFM 47.0412.01)]). 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.
For the entire collection see [Zbl 1234.81013].

MSC:

81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis
81U15	Exactly and quasi-solvable systems arising in quantum theory
14K12	Subvarieties of abelian varieties
12H05	Differential algebra
34L40	Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)

Keywords:

Schrödinger equation; finite-gap potentials; Picard-Vessiot theory; quadratures; Liouvillian extensions; Abelian integrals; theta-functions