Abstract:

This talk concerns the question of completeness for families of eigenfunctions associated to non-linear eigenvalue problems. After presenting the general setting, I will comment on several directions of progress about this question for a few model equations on a segment. These include versions of the non-linear Laplacian eigenvalue problem, the non-linear SchrÃ¶dinger and perhaps a couple of other artificial, but interesting, cases. During the talk it will become evident that the question of completeness is intimately related with deep results about the basis properties of dilated periodic functions. Some of these date back to the pioneering work of Beurling in the mid 1950s and a remarkable framework developed by Hedenmalm, Lindqvist and Seip in the 1990s.