Date of Award
Spring 2015
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics
First Advisor
Rodrigo Banuelos
Committee Chair
Rodrigo Banuelos
Committee Member 1
Fabrice Baudoin
Committee Member 2
Burgess Davis
Committee Member 3
Antonio Sa Barreto
Abstract
Let V be a bounded and integrable potential over Rd and 0 < α ≤ 2. We show the existence of an asymptotic expansion by means of Fourier Transform techniques and probabilistic methods for the following quantities [special characters omitted] and [special characters omitted] as t ↓ 0. These quantities are called the heat trace and heat content in Rd with respect to V, respectively. Here, p((α)/ t)(x, y) and p( HV/t)(x, y) denote, respectively, the heat kernels of the heat semigroups with infinitesimal generators given by (-Δ)(α/2) and HV = (-Δ)(α/2) + V. The former operator is known as the fractional Laplacian whereas the latter one is known as the fractional Schrödinger Operator. ^ The study of the small time behaviour of the above quantities is motivated by the asymptotic expansion as t ↓ 0 of the following spectral functions for smooth bounded domains Ω ⊂ R d, [special characters omitted] where p(Ω,α/ t)(x, y) is the transition density of a stable process killed upon exiting Ω. ^ The function Z((α)/Ω)/)(t) is known as the heat trace and a second order expansion is provided in [6] for all 0 < α ≤ 2 forR-smooth boundary domains. In [5] the result is extended to bounded domains with Lipschitz boundary. As for the spectral function Q((α)/Ω)( t), it is called the spectral heat content and has only been widely studied for the Brownian motion case. In fact, a third order asymptotic expansion is provided in [12] for α = 2. In this work, we will state a conjecture about the second order small time expansion. These expansions differ accordingly to the ranges 1 < α < 2, α = 1 and 0 < α < 1.
Recommended Citation
Valverde, Luis Guillermo Acuna, "Heat trace and heat content asymptotics for Schrodinger Operators of stable processes/fractional Laplacians" (2015). Open Access Dissertations. 577.
https://docs.lib.purdue.edu/open_access_dissertations/577