Harmonic analysis in non-Euclidean geometry: trace formulae and integral representations

Doctoral thesis English OPEN
Awonusika, Richard Olu (2016)
  • Subject: QA440

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  • References (68)
    68 references, page 1 of 7

    1 Introduction and Overview 1 1.1 Historical Background of Non-Euclidean Geometries . . . . . . . . . . . . . . . . 1 1.2 Riemannian Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 The Heat Kernels on Compact Symmetric Spaces . . . . . . . . . . . . . . . . . . 4 1.4 Eigenfunctions of the Laplacian on the Sphere . . . . . . . . . . . . . . . . . . . . 7 1.5 The Upper Half-Space Model of the Hyperbolic Space . . . . . . . . . . . . . . . 14 1.5.1 Eigenfunctions of the Laplacian in Cartesian Coordinates . . . . . . . . . 16 1.5.2 Eigenfunctions of the Laplacian in Geodesic Polar Coordinates . . . . . . 17 1.5.3 The Heat Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5.4 The Resolvent Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5.5 The Wave Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6 The Poincare Unit Ball Model of the Hyperbolic Space . . . . . . . . . . . . . . . 23 1.7 The Poisson Summation Formula as a Trace Formula . . . . . . . . . . . . . . . . 24 1.8 The Selberg Trace Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.9 Outline and Organisation of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 27

    2 The Spectrum and Geometry of Hyperbolic Surfaces 33 2.1 The Upper Half-Plane and the Group SL(2; R) . . . . . . . . . . . . . . . . . . . 34 2.1.1 Classi cations of Isometries of the Upper Half-Plane . . . . . . . . . . . . 35 2.1.2 The Iwasawa Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.1.3 Hyperbolic Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2 Automorphic Forms for the Modular Group SL(2; Z) . . . . . . . . . . . . . . . . 47 2.2.1 The Fourier Expansion of Nonholomorphic Eisenstein Series . . . . . . . . 50 2.2.2 Analytic Continuation and Functional Equation of E(z; s) . . . . . . . . . 53

    3 Trace Formulae for Hyperbolic Surfaces and Applications 56 3.1 The Trace Formula for a Compact Hyperbolic Surface . . . . . . . . . . . . . . . 57 3.1.1 Computation of the Trace for the Identity Element . . . . . . . . . . . . . 62 3.1.2 Computation of the Trace for the Hyperbolic Element . . . . . . . . . . . 62 3.2 The Trace Formula for a Noncompact Hyperbolic Surface . . . . . . . . . . . . . 64 3.2.1 Selberg Spectral Expansion of Automorphic Functions . . . . . . . . . . . 65 3.2.2 The Maass-Selberg Relation . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.2.3 Computation of the Spectral Trace: The Continuous Spectrum . . . . . . 68 3.2.4 Computation of the Trace for the Parabolic Elements . . . . . . . . . . . 70 3.3 The Parseval Inner Product Formula . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.4 Zeta Functions and Determinant of the Laplacian . . . . . . . . . . . . . . . . . . 83 3.4.1 The Heat Trace and Eigenvalue Asymptotics . . . . . . . . . . . . . . . . 83 3.4.2 The Trace of the Resolvent and Selberg Zeta Functions . . . . . . . . . . 87 3.4.3 Zeta Regularised Determinant of the Laplacian . . . . . . . . . . . . . . . 92

    4 Poisson Integral Representations in Euclidean and Non-Euclidean spaces 100 4.1 Eigenfunctions of the Laplacian in the Hyperbolic Space . . . . . . . . . . . . . . 101 4.2 Special Functions Representation of the Poisson Kernel . . . . . . . . . . . . . . 103 4.3 The Poisson Integral Formula for Sn . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.4 Integral Representations of the Euclidean Poisson Kernel . . . . . . . . . . . . . 108 4.5 Series Representations of the Poisson Kernel on Dn . . . . . . . . . . . . . . . . . 117 4.6 Summation of Certain Series Involving Legendre Polynomials . . . . . . . . . . . 121

    5 The Gegenbauer Transform and Heat Kernels on Sn and CPn 126 5.1 The Gegenbauer Transform and its Inversion Formula . . . . . . . . . . . . . . . 126 5.2 The Heat Kernel on Sn via the Gegenbauer Transform . . . . . . . . . . . . . . . 128 5.3 Minakshisundaram-Pleijel Heat Coe cients . . . . . . . . . . . . . . . . . . . . . 134 5.4 Integral Representations of the Heat Kernels on CPn . . . . . . . . . . . . . . . 144 5.5 The Heat Trace Formulae via the Euclidean Poisson Kernel . . . . . . . . . . . . 149

    1 2 [Cm (x)]2 dx = 1 2 Ck (x) dx = 0; k > 0;

    [17] Baker, A. (2012). Matrix Groups: An Introduction to Lie Group Theory. Springer.

    [18] Barnes, E. W. (1901). The theory of the double gamma function. Phil. Trans. Royal Soc. London, 196:265{387.

    [19] Bateman, H., Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. (1953a). Higher Transcendental Functions, Vol. I. McGraw-Hill, New York.

    [20] Bateman, H., Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. (1953b). Higher Transcendental Functions, Vol. II. McGraw-Hill, New York.

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