Brownian Motion on the Hyperbolic Plane and Selberg Trace Formula

Abstract We will show that the relation of the heat kernels for the Schrodinger operators with uniform magnetic fields on the hyperbolic plane H 2 (the Maass Laplacians) and for the Schrodinger operators with Morse potentials on R is given by means of a one-dimensional Fourier transform in the framework of stochastic analysis, where the Brownian motion on H 2 plays an important role. By using this relation, we will give the explicit forms of the Green functions. As a typical related problem, we will discuss the Selberg trace formula. The close relation of the trace formula with the corresponding classical mechanics will also be discussed.

[1]  P. Morse.  Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels , 1929 .

[2]  H. Petersson.  Zur analytischen Theorie der Grenzkreisgruppen , 1938 .

[3]  J. Shohat,et al.  The problem of moments , 1943 .

[4]  M. Kac.  On Some Connections between Probability Theory and Differential and Integral Equations , 1951 .

[5]  A. Selberg.  Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series , 1956 .

[6]  H. McKean.  Elementary solutions for certain parabolic partial differential equations , 1956 .

[7]  L. Landau.  Quantum Mechanics-Nonrelativistic Theory , 1958 .

[8]  V. Vasil’ev,et al.  Waveguides with Random Inhomogeneities and Brownian Motion In the Lobachevsky Plane , 1959 .

[9]  Paul Malliavin,et al.  Stochastic Analysis , 1997, Nature.

[10]  R. A. Silverman,et al.  Special functions and their applications , 1966 .

[11]  W. Roelcke.  Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. II , 1966 .

[12]  H. Buchholz.  The Confluent Hypergeometric Function , 2021, A Course of Modern Analysis.

[13]  H. P. McKean,et al.  Selberg's trace formula as applied to a compact riemann surface , 1972 .

[14]  Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. Teil I , 1973 .

[15]  J. Elstrodt.  Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. Teil II , 1973 .

[16]  Stanislav Molchanov,et al.  DIFFUSION PROCESSES AND RIEMANNIAN GEOMETRY , 1975 .

[17]  S. Patterson.  The laplacian operator on a Riemann surface, III , 1976 .

[18]  John D. Fay.  Fourier coefficients of the resolvent for a Fuchsian group. , 1977 .

[19]  P. Malliavin.  Stochastic calculus of variation and hypoelliptic operators , 1978 .

[20]  N. Ikeda,et al.  Integral of Differential Forms along the Path of Diffusion Processes , 1979 .

[21]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[22]  P. Bougerol.  Exemples de théorèmes locaux sur les groupes résolubles , 1983 .

[23]  Ì. H. Duru.  Morse-potential Green's function with path integrals , 1983 .

[24]  P. Cai,et al.  Path-integral treatment of the Morse oscillator , 1983 .

[25]  S. Patterson.  THE SELBERG TRACE FORMULA FOR PSL(2, R) Volume 2: (Lecture Notes in Mathematics, 1001) , 1984 .

[26]  A. Comtet,et al.  Effective action on the hyperbolic plane in a constant external field , 1985 .

[27]  A. Terras.  Harmonic Analysis on Symmetric Spaces and Applications I , 1985 .

[28]  E. D'hoker,et al.  On determinants of Laplacians on Riemann surfaces , 1986 .

[29]  J. Fischer.  An Approach to the Selberg Trace Formula via the Selberg Zeta-Function , 1987 .

[30]  Peter Sarnak,et al.  Determinants of Laplacians , 1987 .

[31]  A. Comtet.  On the Landau Levels on the Hyperbolic Plane , 1987 .

[32]  Shinzo Watanabe.  Analysis of Wiener Functionals (Malliavin Calculus) and its Applications to Heat Kernels , 1987 .

[33]  Analysis on root systems , 1987 .

[34]  E. D'hoker,et al.  The Geometry of String Perturbation Theory , 1988 .

[35]  C. Grosche.  The path integral on the Poincaré upper half-plane with a magnetic field and for the Morse potential , 1988 .

[36]  Completeness relations for Maass Laplacians and heat kernels on the super Poincaré upper half‐plane , 1990 .

[37]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[38]  Jorge V. José,et al.  Chaos in classical and quantum mechanics , 1990 .

[39]  M. Yor.  On some exponential functionals of Brownian motion , 1992, Advances in Applied Probability.

[40]  Georgeot,et al.  Trace formula for Riemann surfaces with magnetic field. , 1993, Physical review letters.

[41]  A. Comtet,et al.  On the flux distribution in a one dimensional disordered system , 1994 .

[42]  J. Gruet.  Semi-groupe du mouvement brownienhyperbolique , 1996 .

[43]  Path Integrals, Hyperbolic Spaces and Selberg Trace Formulae , 1996 .

[44]  A. Comtet,et al.  Diffusion in a one-dimensional random medium and hyperbolic Brownian motion , 1995, cond-mat/9506024.

[45]  B. Leblanc.  Une approche unifiée pour une forme exacte du prix d'une option dans les différents modèles a volatilité stochastique , 1996 .

[46]  H. McKean,et al.  Diffusion processes and their sample paths , 1996 .

[47]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.