#### 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 **R*** ^{d}* 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

**R**

*with respect to*

^{d}*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

*H*= (-Δ)(α/2) +

_{V}*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**

*, [special characters omitted] where*

^{ d}*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 for

*R*-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