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.
","abstract_html":"<p>Let <em>V</em> be a bounded and integrable potential over <strong>R</strong><em><sup>d</sup></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><sup>d</sup></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><sup> H<sub>V</sub></sup></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<sub>V</sub></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><sup> d</sup></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>","abstract_has_math":false,"creators":["Valverde, Luis Guillermo Acuna"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Rodrigo Banuelos","Fabrice Baudoin","Burgess Davis","Antonio Sa Barreto"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-04-01T07:00:00Z","date_published":"2015-04-01T07:00:00Z","updated_at":"2026-07-24T03:53:41Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://docs.lib.purdue.edu/open_access_dissertations/577","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Rodrigo Banuelos","Fabrice Baudoin","Burgess Davis","Antonio Sa Barreto"]},{"key":"dc:creator","label":"Author","values":["Valverde, Luis Guillermo Acuna"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://docs.lib.purdue.edu/open_access_dissertations/577"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Heat trace and heat content asymptotics for Schrodinger Operators of stable processes/fractional Laplacians"]}]}],"canonical_facts":{"dc:contributor":["Rodrigo Banuelos","Fabrice Baudoin","Burgess Davis","Antonio Sa Barreto"],"dc:creator":["Valverde, Luis Guillermo Acuna"],"dc:description.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.
"],"dc:identifier":["https://docs.lib.purdue.edu/open_access_dissertations/577"],"dc:subject":["Mathematics"],"dc:title":["Heat trace and heat content asymptotics for Schrodinger Operators of stable processes/fractional Laplacians"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:53:41Z"}