Irregular points of modulation

AbstractThe local theory of singular points is extended to a large class of linear, second-order, ordinary differential equations which can be physical Schroedinger equations, or govern the modulation of real oscillators or waves. In addition to Langer's fractional turning points, such equations admit highly irregular points at which the coefficients of the differential equation can be almost arbitrarily multivalued. (Genuine coalescence of singular points, however, is not considered.) A local representation of the solutions is established, which generalizes Frobenius' method of power series, and reveals a remarkable, two-variable structure. Bounds are obtained on the departure of solution structure from the structure characteristic of regular points