We introduce empirical measures to study the $L^2$ norms of restrictions of quantum integrable eigenfunctions to the unique rotationally invariant geodesic $H$ on a convex surface of revolution. The weak* limit of these measures describes the dependence of their size on $H$ in terms of the angular momentum. The limit measures blow up $(1-c^2)^{-1/2}$ at the end points $c = \pm 1$, reflecting the fact that the Gaussian beam sequence is the largest on $H$. We then use the quantized action operators constructed by Colin de Verdière on these surfaces to show that there is a unitary Fourier integral operator which conjugates them to the standard action operators on the round sphere up to finite rank error, showing that all of these surfaces are essentially equivalent in terms of quantum integrability of the Laplacian. Afterwards we move on to study asymptotics of ladder sequences of spherical harmonics and show that they have Airy-type behavior in a shrinking neighborhood of these circles. This provides a more explicit calculation of the quantities appearing in classical expansions by Thorne and Olver for the Legendre functions in terms of the geometry on the sphere. We also include expository notes which are intended to be a practical introduction to homogeneous Lagrangian distributions, Fourier integral operators, and the symbol calculus of composition. The primary focus is on examples and line-by-line calculation, including a description of the singularities of the Duistermaat-Guillemin wave trace using the symbol calculus.

Creator

• Geis, Michael Leamy

DOI

• https://doi.org/10.21985/n2-j190-0h41

Subject

• Mathematics

Language

• en

Alternate Identifier

• http://dissertations.umi.com/northwestern:16276
• etdadmin_upload_929536

Keyword

• convex surface of revolution
• asymptotics
• eigenfunctions
• empirical measure
• spectrum

Date created

• 2022-01-01

Resource type

• Dissertation

Rights statement

• In Copyright