Heat trace and heat content asymptotics for Schrodinger Operators of stable processes/fractional Laplacians

<p>Let <em>V</em> be a bounded and integrable potential over <strong>R</strong><em>^d</em> 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 <em>t</em> ↓ 0. These quantities are called the <em>heat trace</em> and <em>heat content</em> in <strong>R</strong><em>^d</em> with respect to <em>V</em>, respectively. Here, <em>p</em>((α)/<em> t</em>)(<em>x, y</em>) and <em>p</em>(<em>^H_V</em>/<em>t</em>)(<em>x, y</em>) denote, respectively, the heat kernels of the heat semigroups with infinitesimal generators given by (-Δ)(α/2) and <em>H_V</em> = (-Δ)(α/2) + <em>V</em>. 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 <em>t</em> ↓ 0 of the following spectral functions for smooth bounded domains Ω ⊂ <strong>R</strong><em>^d</em>, [special characters omitted] where <em>p</em>(Ω,α/<em> t</em>)(<em>x, y</em>) is the transition density of a stable process killed upon exiting Ω. ^ The function <em>Z</em>((α)/Ω)/)(<em>t</em>) is known as the heat trace and a second order expansion is provided in [6] for all 0 < α ≤ 2 for<em>R</em>-smooth boundary domains. In [5] the result is extended to bounded domains with Lipschitz boundary. As for the spectral function <em>Q</em>((α)/Ω)(<em> t</em>), it is called the <em>spectral heat content</em> 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.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2015

Author dc:creator

• Valverde, Luis Guillermo Acuna

Contributors dc:contributor

• Rodrigo Banuelos
• Fabrice Baudoin
• Burgess Davis
• Antonio Sa Barreto

• Mathematics

Repository record dc:identifier

https://docs.lib.purdue.edu/open_access_dissertations/577

OAI identifier oai:identifier

oai:docs.lib.purdue.edu:open_access_dissertations-1546