NDLI: Courant-sharp eigenvalues of Neumann 2-rep-tiles

Content Provider	Springer Nature Link
Author	Bersudsky, Michael Band, Ram Fajman, David
Copyright Year	2016
Abstract	We find the Courant-sharp Neumann eigenvalues of the Laplacian on some 2-rep-tile domains. In $$\mathbb {R}^{2}$$ , the domains we consider are the isosceles right triangle and the rectangle with edge ratio $$\sqrt{2}$$ (also known as the A4 paper). In $$\mathbb {R}^{n}$$ , the domains are boxes which generalize the mentioned planar rectangle. The symmetries of those domains reveal a special structure of their eigenfunctions, which we call folding\unfolding. This structure affects the nodal set of the eigenfunctions, which, in turn, allows to derive necessary conditions for Courant-sharpness. In addition, the eigenvalues of these domains are arranged as a lattice which allows for a comparison between the nodal count and the spectral position. The Courant-sharpness of most eigenvalues is ruled out using those methods. In addition, this analysis allows to estimate the nodal deficiency—the difference between the spectral position and the nodal count.
Ending Page	859
Page Count	39
Starting Page	821
File Format	PDF
ISSN	03779017
e-ISSN	15730530
Journal	Letters in Mathematical Physics
Issue Number	5
Volume Number	107
Language	English
Publisher	Springer Netherlands
Publisher Date	2016-11-29
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Geometry Courant nodal theorem Nodal set Theoretical, Mathematical and Computational Physics Complex Systems Spectral theory; eigenvalue problems Oscillation, zeros of solutions, mean value theorems, etc. Nodal domains Group Theory and Generalizations Spectral problems; spectral geometry; scattering theory
Content Type	Text
Resource Type	Article
Subject	Statistical and Nonlinear Physics Mathematical Physics

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