Title: Quasimodes for generalized semi-classical differential operators, Newton polygons, and blow-ups
Abstract: We consider families Ph of linear differential operators on an interval that depend on a parameter h≥0 and degenerate as h→ 0. We consider the problem of constructing quasimodes, i.e. (families of) solutions uh, h>0, of Ph uh = O(h∞) as h→ 0. A classical example is the semi-classical Schroedinger Operator Ph = h2 ∂2 + V where ∂ = d/dx and V is a smooth function. If V is positive then quasimodes can be found using the standard WKB method. At zeroes of V additional difficulties arise (solved by Olver long ago) due to different scaling behavior near and away from the zeroes. Another classical example is Bessel's equation with parameter \nu=1/h, where the behavior of solutions uniformly for large parameter and large argument is of interest (and well-known). One application of quasimode constructions is to find approximations of the spectrum of Ph for small positive h.
We construct, and give a precise description of, full sets of quasimodes for a very large class of families Ph=P(x,∂x,h) of any order, which includes almost all families where the coefficients depend analytically on x and h, under a mild genericity hypothesis. It also includes the examples above. The generality of the setup leads to a high degree of combinatorial and analytic complexity, which can be handled by an efficient representation of the data by Newton polygons and of the result in terms of iterated blow-ups and a suitable class of oscillatory-polyhomogeneous functions.
This is joint work with Dennis Sobotta.
