Picard Operators

A linear differential expression Ly = y (n) + qn ‐2 y n‐2 +…+ q 0 y is called a Picard expression if its coefficients are elliptic functions (with common fundamental periods) and if the general solution of Ly = Ey is an everywhere meromorphic function (with respect to the independent variable) for all E ∈ ℂ. If L is a Picard expression we show that the differential equation Ly = Ey has n linearly independent Floquet solutions except when E is any of a finite number of exceptional values. Also the conditional stability set of a Picard expression (and hence the spectrum of the associated operator in L 2 (ℝ)) consists of finitely many regular analytic arcs.