Several results including integral representation of solutions and
Hermite-Krichever Ansatz on Heun's equation are generalized to a certain class
of Fuchsian differential equations, and they are applied to equations which are
related with physics. We investigate linear differential equations that produce
Painlev\'e equation by monodromy preserving deformation and obtain solutions of
the sixth Painlev\'e equation which include Hitchin's solution. The
relationship with finite-gap potential is also discussed. We find new
finite-gap potentials. Namely, we show that the potential which is written as
the sum of the Treibich-Verdier potential and additional apparent singularities
of exponents -1 and 2 is finite-gap, which extends the result obtained
previously by Treibich. We also investigate the eigenfunctions and their
monodromy of the Schr\"odinger operator on our potential.
Publié le : 2005-04-26
Classification:
Mathematics - Classical Analysis and ODEs,
Mathematical Physics,
Mathematics - Quantum Algebra,
Nonlinear Sciences - Exactly Solvable and Integrable Systems,
82B23,
34M55,
33E10,
33E15
@article{0504540,
author = {Takemura, Kouichi},
title = {The Hermite-Krichever ansatz for Fuchsian equations with applications to
the sixth Painlev\'e equation and to finite-gap potentials},
journal = {arXiv},
volume = {2005},
number = {0},
year = {2005},
language = {en},
url = {http://dml.mathdoc.fr/item/0504540}
}
Takemura, Kouichi. The Hermite-Krichever ansatz for Fuchsian equations with applications to
the sixth Painlev\'e equation and to finite-gap potentials. arXiv, Tome 2005 (2005) no. 0, . http://gdmltest.u-ga.fr/item/0504540/