Ubiquitous Heat Kernel.

Intro -- Contents -- Preface -- Positivity of zeta distributions and small unitary representations -- The heat equation and representations of the Jacobi group -- Kato's inequality and asymptotic spectral properties for discrete magnetic Laplacians -- The heat kernel in low-dimensional quantum theories -- Heat kernels on weighted manifolds and applications -- 1. Introduction -- 2. The Laplace operator -- 2.1 Differential operators on manifolds -- 2.2 Laplacian as an operator in L2 -- 2.3 Some examples -- 2.4 Laplacian on model manifolds -- 3. The heat kernel -- 3.1 Heat semigroup -- 3.2 Heat kernel and fundamental solutions -- 3.3 Stochastic completeness -- 4. Relations between different heat kernels -- 4.1 Direct products -- 4.2 Isometries -- 4.3 Comparison of heat kernels -- 4.4 Change of measure -- 4.5 Some examples of heat kernels in R -- 4.6 Hyperbolic spaces -- 5. Heat kernel estimates -- 5.1 Uniform Faber-Krahn inequality -- 5.2 Gaussian upper bounds -- 5.3 On-diagonal lower bounds -- 5.4 Relative Faber-Krahn inequality -- 5.5 On-diagonal estimates on model manifolds -- 5.6 Estimates with the mean curvature function -- 5.7 Green function and Green operator -- 6. Harnack inequality -- 6.1 The Li-Yau estimate -- 6.2 Manifolds with relatively connected annuli -- 6.3 Non-uniform change of measure -- 6.4 Conformal change of the metric tensor -- 6.5 Manifolds with ends -- 7. Eigenvalues of Schrödinger operators -- 7.1 Negative eigenvalues -- 7.2 Stability index of minimal surfaces -- 8. The Brownian motion -- 8.1 Construction of the Brownian motion -- 8.2 The first exit time -- 8.3 The Dirichlet problem for a Schrödinger operator -- 9. Path properties of stochastic processes -- 9.1 Recurrence and transience -- 9.2 Escape rate -- 9.3 Recurrence and transience of α-process -- 9.4 Asymptotic separation of trajectories of α-process.

10. Heat kernels of Schrödinger operators -- 10.1 Heat kernel and a ground state -- 10.2 Green bounded potentials -- 10.3 Potentials with a polynomial ground state -- 10.4 Potentials of quadratic decay in Rn -- 10.5 Spherically symmetric potentials -- 10.6 Appendix: behavior of harmonic functions at ∞ -- References -- Heat kernels in geometric evolution equations -- The range of the heat operator -- Heat kernels and cycles -- Green currents on Kähler manifolds -- Heat kernels and Riesz transforms -- The contest between the kernels in the Selberg trace formula for the (q + 1)-regular tree -- Expressing Arakelov invariants using hyperbolic heat kernels -- Grassmanians of higher local fields and multivariable tau functions -- Heat kernels and the range of the trace on completions of twisted group algebras -- Theta functions, old and new -- The heat kernel on the symmetric space SL(n, F)/SU(n, F) -- 1. Introduction -- 2. The root system An-1 with generalized root multiplicities -- 3. The space SL(n, R)/SO(n) -- 4. The root system An-1 with even root multiplicities -- 5. The space SL(n, C)/SU(n) -- 6. The space SL(n, H)/Sp(n) -- Conclusion -- References -- Incidence structure.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.