The Hermite-Krichever ansatz for Fuchsian equations with applications to the sixth Painlev\'e equation and to finite-gap potentials

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.