Analysis Seminar

Title: Quasimodes for generalized semi-classical differential operators, Newton polygons, and blow-ups
Speaker: Daniel Grieser
Speaker Info: Oldenburg
Brief Description:
Special Note: Note special day, time, location.

We consider families $P_h$ of differential operators on an interval that depend on a parameter $h\geq 0$ and degenerate as $h\to 0$. We consider the problem of constructing quasimodes, i.e. (families of) solutions $u_h$, $h>0$, of $P_h u_h = O(h^\infty)$ as $h\to 0$. A classical example is the semi-classical Schroedinger Operator $P_h = h^2 \partial^2 + V$ where $\partial = d/dx$ 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).

We construct, and give a precise description of, full sets of quasimodes for very general families $P_h=P(x,\partial_x,h)$ of any order, including the examples above as well as operators where the coefficients depend analytically on $x$ and $h$, under a mild genericity hypothesis. 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.

Date: Tuesday, May 21, 2024
Time: 4:00pm
Where: Lunt 107
Contact Person: Prof. Jared Wunsch
Contact email: jwunsch@math.northwestern.edu
Contact Phone: 847-491-5580
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