Floquet theory for doubly periodic differential equations.

Our objective is to extend the well-known Floquet theory of ordinary differential equations with singly periodic coefficients, to equations with doubly-periodic coefficients. We study mainly an equation of fairly general type, analogous to Hill's equation, hut doubly-periodic. Some particular attention is devoted, however, to the special case of Lame's equation. A general theory, analogous to that for Hill's equation, is first developed, with some consideration of an algebraic form of the equation, having three regular singularities and one irregular. Next we introduce a parameter v (one of the characteristic exponents at a singularity). In the case v = O the general solution is uniform and Hermite showed that there then exists at least one doubly-multiplicative solution. The central work of this, thesis is to consider certain rational values of V, introducing some special cuts in the complex plane and showing that in certain circumstances the general solution is uniform in the cut plane. When this is so, doubly-multiplicative solutions again exist. Extension to general rational values of v depends on an interesting and apparently unproved conjecture related to the zeros of Chebyshev polynomials