By Werner Müller (auth.)
ISBN-10: 3540176969
ISBN-13: 9783540176961
ISBN-10: 3540477624
ISBN-13: 9783540477624
The manifolds investigated during this monograph are generalizations of (XX)-rank one in the neighborhood symmetric areas. within the first a part of the ebook the writer develops spectral concept for the differential Laplacian operator linked to the so-called generalized Dirac operators on manifolds with cusps of rank one. This comprises the case of spinor Laplacians on (XX)-rank one in the neighborhood symmetric areas. The time-dependent method of scattering thought is taken to derive the most effects concerning the spectral answer of those operators. the second one a part of the booklet offers with the derivation of an index formulation for generalized Dirac operators on manifolds with cusps of rank one. This index formulation is used to turn out a conjecture of Hirzebruch about the relation of signature defects of cusps of Hilbert modular forms and specific values of L-series. This publication is meant for readers operating within the box of automorphic varieties and research on non-compact Riemannian manifolds, and assumes an information of PDE, scattering thought and harmonic research on semisimple Lie groups.
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Additional info for Manifolds with Cusps of Rank One: Spectral Theory and L2-Index Theorem
Sample text
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To following : a for which 6 ]R I},l < and f 6 C o ( 1 9 - { a }) c . Then, depends on f for and every m . Choose m , such s > O 6 iN , t h e r e that, for such that exists a t 6 II - {O} 51 and lYl < ~ I t l , one has If e 2 i y X + i t X 2 f ( X 2+a) d Xl < C l t l -m O PROOF. D. Let - If we > ~/z . 35) co and the k=l can be e s t i m a t e d by C l t ! 12) oo ~ 2-~(f n(r,~ Ix[ < c / 2 . 11) L2(FM\XM'EM ) (e-itHoa(Ho)~)(r,x) = Then IXl < ~ - i¢)e - i t ( ¢ 2 + m2/4 + ~ k ) a ( # + m2/4 + ~k )" o • (J~)(c,k)dc))~ k(X) Let co= P v , vE L 2 ( [ c , co) XFM\XM,EM) and p u t w = F*Z*V*Jv + O O wE L2(IR; 2) and J~ = VZ+Fo(X+W) If we use the definition of and F ° , then we get (J~)(¢,k) = 2/~¢ lim X ÷e° ~ e-i(¢Z+m2/4 +~k)Sw(s,k) Then V,Z+ ds , O where the limit is taken in L2 Assume that t > 0.
Manifolds with Cusps of Rank One: Spectral Theory and L2-Index Theorem by Werner Müller (auth.)
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