# 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.

**DOI: **http://dx.doi.org/10.7900/jot.2020jun08.2304

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

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