Asymptotic expansion of polyanalytic Bergman kernels
2014 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 267, no 12, 4667-4731 p.Article in journal (Refereed) Published
We consider the q-analytic functions on a given planar domain Omega, square integrable with respect to a weight. This gives us a q-analytic Bergman kernel, which we use to extend the Bergman metric to this context. We recall that f is q-analytic if (partial derivative) over bar (q) f = 0 for the given positive integer q. Polyanalytic Bergman spaces and kernels appear naturally in time-frequency analysis of Gabor systems of Hermite functions as well as in the mathematical physics of the analysis of Landau levels.
We obtain asymptotic formulae in the bulk for the q-analytic Bergman kernel in the setting of the power weights e(-2mQ), as the positive real parameter m tends to infinity. This is only known previously for q = 1, by the work of Tian, Yau, Zelditch, and Catlin. Our analysis, however, is inspired by the more recent approach of Berman, Berndtsson, and Sjostrand, which is based on ideas from microlocal analysis.
We remark here that since a q-analytic function may be identified with a vector-valued holomorphic function, the Bergman space of q-analytic functions may be understood as a vector-valued holomorphic Bergman space supplied with a certain singular local metric on the vectors. Finally, we apply the obtained asymptotics for q = 2 to the bianalytic Bergman metrics, and after suitable blow-up, the result is independent of Q for a wide class of potentials Q. We interpret this as an instance of geometric universality.
Place, publisher, year, edition, pages
2014. Vol. 267, no 12, 4667-4731 p.
Polyanalytic functions, Bergman kernel, Asymptotic expansion, Bulk universality
IdentifiersURN: urn:nbn:se:kth:diva-122142DOI: 10.1016/j.jfa.2014.09.002ISI: 000346226500003ScopusID: 2-s2.0-84921969223OAI: oai:DiVA.org:kth-122142DiVA: diva2:620977
FunderGöran Gustafsson Foundation for Research in Natural Sciences and MedicineSwedish Research Council, 2012-3122
QC 20150220. Updated from submitted to published.2013-05-132013-05-132015-02-20Bibliographically approved