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Sums of Reciprocal Eigenvalues of the Laplacian
Author(s) -
Dittmar Bodo
Publication year - 2002
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200204)237:1<45::aid-mana45>3.0.co;2-m
Subject(s) - mathematics , reciprocal , conformal map , eigenvalues and eigenvectors , complex plane , plane (geometry) , unit disk , unit (ring theory) , laplace operator , radius , mathematical analysis , spectral radius , simply connected space , combinatorics , pure mathematics , geometry , quantum mechanics , physics , linguistics , mathematics education , computer security , computer science , philosophy
The purpose of this paper is to present formulas for the sum of the squares of all reciprocal eigenvalues λ ofthe fixed membrane and the free membrane μ for plane simply connected domains. As a consequence we obtain that among all simply connected plane domains with the maximal conformal radius 1 and the area A only the unit disk yields the minimum of$$ A^{2} \sum _{2} ^{\infty} {1 \over \mu ^{2} _{j}} $$ .