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.

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Committee Chair

Rodrigo Banuelos

Date of Award

Spring 2015

First Advisor

Rodrigo Banuelos

Committee Member 1

Fabrice Baudoin

Committee Member 2

Burgess Davis

Committee Member 3

Antonio Sa Barreto

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