The worthy purpose of this text is to provide a complete, self-contained development of the trace formula and theta inversion formula for SL(2,Z[i])\SL(2,C). Unlike other treatments of the theory, the approach taken here is to begin with the heat kernel on SL(2,C) associated to the invariant Laplacian, which is derived using spherical inversion. The heat kernel on the quotient space SL(2,Z[i])\SL(2,C) is arrived at through periodization, and further expanded in an eigenfunction expansion. A theta inversion formula is obtained by studying the trace of the heat kernel. Following the author's previous work, the inversion formula then leads to zeta functions through the Gauss transform.

The Heat Kernel and Theta Inversion on SL2(C) (Springer Monographs in Mathematics)

Science & Math

Jorgenson, Jay

Lang, Serge

Mathematics

Springer

For the most part the authors are concerned with SLn(R) and with invariant differential operators, the invarinace being with respect to various subgroups. To a large extent, this book carries out the general results of Harish-Chandra.

Spherical Inversion on SLn(R) (Springer Monographs in Mathematics)

In this volume, the authors study asymptotics of the geometry and spectral theory of degenerating sequences of finite volume hyperbolic manifolds of three dimensions. Thurston's hyperbolic surgery theorem asserts the existence of non-trivial sequences of finite volume hyperbolic three manifolds which converge to a three manifold with additional cusps. In the geometric aspect of their study, the authors use the convergence of hyperbolic metrics on the thick parts of the manifolds under consideration to investigate convergence of tubes in the manifolds of the sequence to cusps of the limiting manifold. 
In the spectral theory aspect of the work, they prove convergence of heat kernels. They then define a regularized heat trace associated to any finite volume, complete, hyperbolic three manifold, and study its asymptotic behavior through degeneration. As an application of the analysis of the regularized heat trace, they study asymptotic behavior of the spectral zeta function, determinant of the Laplacian, Selberg zeta function, and spectral counting functions through degeneration. 
The authors' methods are an adaptation to three dimensions of the earlier work of Jorgenson and Lundelius who investigated the asymptotic behavior of spectral functions on degenerating families of finite area hyperbolic Riemann surfaces.

Spectral Asymptotics on Degenerating Hyperbolic 3-Manifolds (Memoirs of the American Mathematical Society)

Dodziuk, Jozef

Amer Mathematical Society

The worthy purpose of this text is to provide a complete, self-contained development of the trace formula and theta inversion formula for SL(2,Z[i])\SL(2,C). Unlike other treatments of the theory, the approach taken here is to begin with the heat kernel on SL(2,C) associated to the invariant Laplacian, which is derived using spherical inversion. The heat kernel on the quotient space SL(2,Z[i])\SL(2,C) is arrived at through periodization, and further expanded in an eigenfunction expansion. A theta inversion formula is obtained by studying the trace of the heat kernel. Following the author's previous work, the inversion formula then leads to zeta functions through the Gauss transform.

Posn(R) and Eisenstein Series provides an introduction, requiring minimal prerequisites, to the analysis on symmetric spaces of positive definite real matrices as well as quotients of this space by the unimodular group of integral matrices. The approach is presented in very classical terms and includes material on special functions, notably gamma and Bessel functions, and focuses on certain mathematical aspects of Eisenstein series.

Posn(R) and Eisenstein Series (Lecture Notes in Mathematics, Vol. 1868) (Lecture Notes in Mathematics, 1868)

Analytic number theory and part of the spectral theory of operators (differential, pseudo-differential, elliptic, etc.) are being merged under amore general analytic theory of regularized products of certain sequences satisfying a few basic axioms. The most basic examples consist of the sequence of natural numbers, the sequence of zeros with positive imaginary part of the Riemann zeta function, and the sequence of eigenvalues, say of a positive Laplacian on a compact or certain cases of non-compact manifolds. The resulting theory is applicable to ergodic theory and dynamical systems; to the zeta and L-functions of number theory or representation theory and modular forms; to Selberg-like zeta functions; andto the theory of regularized determinants familiar in physics and other parts of mathematics. Aside from presenting a systematic account of widely scattered results, the theory also provides new results. One part of the theory deals with complex analytic properties, and another part deals with Fourier analysis. Typical examples are given. This LNM provides basic results which are and will be used in further papers, starting with a general formulation of Cram r's theorem and explicit formulas. The exposition is self-contained (except for far-reaching examples), requiring only standard knowledge of analysis.

Basic Analysis of Regularized Series and Products (Lecture Notes in Mathematics, 1564)

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Books by Jorgenson, Jay

See the best price to sell, buy, or rent books by Jorgenson, Jay. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

The Heat Kernel and Theta Inversion on SL2(C) (Springer Monographs in Mathematics)

Spherical Inversion on SLn(R) (Springer Monographs in Mathematics)

Spectral Asymptotics on Degenerating Hyperbolic 3-Manifolds (Memoirs of the American Mathematical Society)

The Heat Kernel and Theta Inversion on SL2(C) (Springer Monographs in Mathematics)

Spherical Inversion on SLn(R) (Springer Monographs in Mathematics)

Posn(R) and Eisenstein Series (Lecture Notes in Mathematics, Vol. 1868) (Lecture Notes in Mathematics, 1868)

Basic Analysis of Regularized Series and Products (Lecture Notes in Mathematics, 1564)