The Weil representation, Maslov index and Theta series, Gerard Lion; Michele Vergne

Автор: Victor P. Maslov; M.A. Shishkova; A.B. Sossinsky
Название: The Complex WKB Method for Nonlinear Equations I
ISBN: 3034896697 ISBN-13(EAN): 9783034896696
Издательство: Springer
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Описание: This book deals with asymptotic solutions of linear and nonlinear equa- tions which decay as h ---+ 0 outside a neighborhood of certain points, curves and surfaces. Such solutions are almost everywhere well approximated by the functions cp(x) exp{iS(x)/h}, x E 1R3, where S(x) is complex, and ImS(x) o. When the phase S(x) is real (ImS(x) = 0), the method for obtaining asymp- totics of this type is known in quantum mechanics as the WKB-method. We preserve this terminology in the case ImS(x) 0 and develop the method for a wide class of problems in mathematical physics. Asymptotics of this type were constructed recently for many linear prob- lems of mathematical physics; certain specific formulas were obtained by differ- ent methods (V. M. Babich 5 -7], V. P. Lazutkin 76], A. A. Sokolov, 1. M. Ter- nov 113], J. Schwinger 107, 108], E. J. Heller 53], G. A. Hagedorn 50, 51], V. N. Bayer, V. M. Katkov 21], N. A. Chernikov 35] and others). However, a general (Hamiltonian) formalism for obtaining asymptotics of this type is clearly required; this state of affairs is expressed both in recent mathematical and physical literature. For example, the editors of the collected volume 106] write in its preface: "One can hope that in the near future a computational pro- cedure for fields with complex phase, similar to the usual one for fields with real phase, will be developed.

Автор: Eberhard Freitag
Название: Singular Modular Forms and Theta Relations
ISBN: 3540547045 ISBN-13(EAN): 9783540547044
Издательство: Springer
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Описание: This research monograph reports on recent work on the theory of singular Siegel modular forms of arbitrary level. Singular modular forms are represented as linear combinations of theta series. The reader is assumed to know only the basic theory of Siegel modular forms.

Автор: Jay Jorgenson; Serge Lang
Название: The Heat Kernel and Theta Inversion on SL2(C)
ISBN: 0387380310 ISBN-13(EAN): 9780387380315
Издательство: Springer
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Описание: Provides the development of the trace formula and theta inversion formula for SL(2,Z[i]) SL(2,C). Unlike other treatments of the theory, this book begins with the heat kernel on SL(2,C) associated to the invariant Laplacian, which is derived using spherical inversion.

Автор: Gelca Razvan
Название: Theta Functions and Knots
ISBN: 9814520578 ISBN-13(EAN): 9789814520577
Издательство: World Scientific Publishing
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Описание:

This book presents the relationship between classical theta functions and knots. It is based on a novel idea of Răzvan Gelca and Alejandro Uribe, which converts Weil's representation of the Heisenberg group on theta functions to a knot theoretical framework, by giving a topological interpretation to a certain induced representation. It also explains how the discrete Fourier transform can be related to 3- and 4-dimensional topology.

Theta Functions and Knots can be read in two perspectives. Readers with an interest in theta functions or knot theory can learn how the two are related. Those interested in Chern-Simons theory will find here an introduction using the simplest case, that of abelian Chern-Simons theory. Moreover, the construction of abelian Chern-Simons theory is based entirely on quantum mechanics and not on quantum field theory as it is usually done.

Both the theory of theta functions and low dimensional topology are presented in detail, in order to underline how deep the connection between these two fundamental mathematical subjects is. Hence the book is self-contained with a unified presentation. It is suitable for an advanced graduate course, as well as for self-study.

Автор: Jay Jorgenson; Serge Lang
Название: The Heat Kernel and Theta Inversion on SL2(C)
ISBN: 1441922822 ISBN-13(EAN): 9781441922823
Издательство: Springer
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Описание: The purpose of this text is to provide a 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, this one begins with the heat kernel on SL(2,C) associated to the invariant Laplacian, which is derived using spherical inversion.

Автор: Shaun Cooper
Название: Ramanujan`s Theta Functions
ISBN: 3319561715 ISBN-13(EAN): 9783319561714
Издательство: Springer
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Описание: Preface.- 0. Sum to Product Identities.- 1. Elliptic Functions.- 2. Transformations.- 3. Theta Functions.- 4. Levels 1, 2, 3, and 4: Jacobi's Inversion Theorem and Ramanujan's Alternative Theories.- 5. Level 5: The Rogers-Ramanujan Continued Fraction.- 6. Level 6: Ramanujan's Cubic Continued Fraction.- 7. Level 7.- 8. Level 8: The Ramanujan-Gollnitz-Gordon Continued Fraction.- 9. Level 9.- 10. Level 10: Ramanujan's Function k.- 11. Levels 11 and 23.- 12. Level 12.- 13. Hypergeometric Modular Transformations.- 14. Ramanujan's Series for 1/pi.- References.