Darboux-Lamé equation and isomonodoromic deformation on the torus

Mayumi Ohmiya Doshisha Universuty

For the n-th Lamé equation

\frac{d^{2}f}{dx^{2}} = (n(n + 1)℘(x, τ) − λ)f,

where ℘(x, τ) is Weierstrass elliptic function with the primitive periods 1 and τ, ℑτ > 0, and n is a natural number, one can construct the double Darboux transformation via some specified eigenfunction, which is the 1-parameter family of ordinary differential equations with the algebro geometric elliptic potentials. It is shown that such 1-parameter family of ordinary differential equations is the isomonodromic family of Fuchsian equations on the torus ℂ/L, where L = ℤ ⊕ ℤτ is the lattice.