Research on the spectrum and the monodromy related to the algebro-geometric potentials

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When the n-th Lame potential (n≧2), which is the most typical algebro-geometric potential, satisfies a kind of the degenerate condition, the Darboux-Lame equations are defined, and studied from the both of the viewpoints of isospectral deformation and isomonodromic deformation. The resulting equations are shown to be isospectral by verifying that those potentials solve the higher order KdV equation.On the one hand, those equations are shown to be isomonodromic if and only if the deformation parameter is restricted to be on some specific algebraic curve. It is also shown that the 2nd Darboux-Lame equation is transformed to the equation of Heun type with the four regular singular points by some 3-fold covering map from the underlying elliptic curve to the protective line. In addition, if the deformation parameter coincides with zero, this Heun type equation degenerates to Gauss' hypergeometric equation with the three regular singular points. By vitue of this fact and its isomonodromic property, the monodromy group associated with this Heun type equation can be concretely computed.Moreover, a kind of fibering condition of the 3-manifolds over the circle is clarified to study the fibering structure of the Jacobi variety corresponding to the spectral curve.To calculate concretely the monodromy representation, it is necessary to clarify the structure of the invariant ring associated with the finete group. An algorithm to compute the generator of the invariant ring is explored and inplemented with the computer algebra system Asir.