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Journal of Operator Theory

Volume 86, Issue 2, Fall 2021  pp. 439-467.

A real analyticity result for symmetric functions of the eigenvalues of a quasiperiodic spectral problem for the Dirichlet Laplacian

Authors:  Massimo Lanza de Cristoforis (1), Paolo Musolino (2), Jari Taskinen (3)
Author institution:(1) Dipartimento di Matematica ``Tullio Levi-Civita'', Universita degli Studi di Padova, Padova, 35121, Italy
(2) Dipartimento di Scienze Molecolari e Nanosistemi, Universita Ca' Foscari Venezia, Venezia Mestre, 30170, Italy
(3) Department of Mathematics and Statistics, University of Helsinki, Helsinki, 00014, Finland

Summary: As is well known, by the Floquet--Bloch theory for periodic problems, one can transform a spectral Laplace--Dirichlet problem in the plane with a set of periodic perforations into a family of ``model problems'' depending on a parameter $\eta \in [0,2\pi]^2$ for quasiperiodic functions in the unit cell with a single perforation. We prove real analyticity results for the eigenvalues of the model problems upon perturbation of the shape of the perforation of the unit~cell.

Keywords: real analytic, domain perturbation, Laplace-Dirichlet problem, periodic domain, Floquet-Bloch theory, band-gap spectrum

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