Download Spectral Theory of Automorphic Functions (Proceedings of the by Alexei B. Venkov PDF

By Alexei B. Venkov

'Et moi, ..., si j'avait su remark en revcnrr, One provider arithmetic has rendered the je n'y serais aspect aile.' human race. It has positioned logic again. Jules Verne the place it belongs, at the topmost shelf subsequent to the dusty canister labelled 'discarded non The sequence is divergent; hence we might be sense'. capable of do whatever with it. Eric T. Bell O. Heaviside arithmetic is a device for proposal. A hugely valuable instrument in a global the place either suggestions and non linearities abound. equally, all types of components of arithmetic function instruments for different elements and for different sciences. utilizing an easy rewriting rule to the quote at the correct above one reveals such statements as: 'One carrier topology has rendered mathematical physics .. .'; 'One carrier good judgment has rendered com puter technology .. .'; 'One carrier class thought has rendered arithmetic .. .'. All arguably precise. And all statements accessible this manner shape a part of the raison d'etre of this sequence.

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Y, s ) = ( lj( z ) exp(- 27TijX ) dx . Jo We compute the coefficients a/ y, s ) : a/ y, s ) = . 11 2: yS( g�l ygp z ) pp x*( y ) e / a ) exp { - 2 7TijX ) dx , 0 y E f,,\ f (3 . 1 . 2 ) j E ll. We introduce notation for the matrix elements of the transformation : g� lygpz = ( az + b )/ ( cz + d ) , c = C( g�lygp ) , d = d ( g� l ygp ) . ( 3 . 1 . 3 ) We have singled out the elements c and d , since they play a special role here. According to the remark following Theorem 1 . 7, for y E r we have c( g� I ygp ) =1= 0, as long as a =1= {3.

8 (s) realizes a meromorphic continuation of the resolvent 9T(s) as an integral operator in the half-plane Re s > O. §2 . 3 . PECTRUM 35 We now explain the behavior of the kernels b(z, z'; s) and r(z, z' ; s) for the operator SB (s) and 9l (s), respectively, in a neighborhood of an arbitrary singular point So for which Re So � 1/2. 26 ) v ; v ; s ) :j,�= I ( z j ) \8) ( z' j) + r(z z' ) ( where {v(z; j )}jl2, I is a real basis of the eigensubspace of X( f; the n o-fold eigenvalue Ao of the operator 21 ; furthermore, X) corresponding to f( z ; j ) = v ( z ; j) - w( so ) � ( K ) v ( z ; j ) ; the kernels 6(z, z'; s) and fez, z ' ; s) are analytic in s in a neighborhood oLso, and (8) denotes the tensor product of the values of functions in V.

1 and Theorems 3. 1 . 2 it follows that the difference £( z; s; IX; etC IX» - 1/;( z; s; IX; I) is bounded in z and is an eigenfunc­ tion for the operator m ( f ; X ) with eigenvalue A = s( 1 - s) for any s, 1 < Re s < 2. Since m (f; X) is a selfadjoint operator, this is only possible if this difference is identically zero. The proof is complete. PROOF. 3. 5 . leromorphically in s onto all of C , 1 � a � n, 1 � I � k 1 � f3 � n, 1 � k � kf3,} E Z, z E H. 2) The kernel of the resolvent 91 (s; f; X) and the kernel of the operator iB (s; f; X) (see Chapter 2) can be extended meromorphically in s onto all of C .

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