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You searched for subject:(Wodzicki residue). Showing records 1 – 2 of 2 total matches.

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Loughborough University

1. Li, Liangpan. Local spectral asymptotics and heat kernel bounds for Dirac and Laplace operators.

Degree: PhD, 2016, Loughborough University

In this dissertation we study non-negative self-adjoint Laplace type operators acting on smooth sections of a vector bundle. First, we assume base manifolds are compact, boundaryless, and Riemannian. We start from the Fourier integral operator representation of half-wave operators, continue with spectral zeta functions, heat and resolvent trace asymptotic expansions, and end with the quantitative Wodzicki residue method. In particular, all of the asymptotic coefficients of the microlocalized spectral counting function can be explicitly given and clearly interpreted. With the auxiliary pseudo-differential operators ranging all smooth endomorphisms of the given bundle, we obtain certain asymptotic estimates about the integral kernel of heat operators. As applications, we study spectral asymptotics of Dirac type operators such as characterizing those for which the second coefficient vanishes. Next, we assume vector bundles are trivial and base manifolds are Euclidean domains, and study non-negative self-adjoint extensions of the Laplace operator which acts component-wise on compactly supported smooth functions. Using finite propagation speed estimates for wave equations and explicit Fourier Tauberian theorems obtained by Yuri Safarov, we establish the principle of not feeling the boundary estimates for the heat kernel of these operators. In particular, the implied constants are independent of self-adjoint extensions. As a by-product, we affirmatively answer a question about upper estimate for the Neumann heat kernel. Finally, we study some specific values of the spectral zeta function of two-dimensional Dirichlet Laplacians such as spectral determinant and Casimir energy. For numerical purposes we substantially improve the short-time Dirichlet heat trace asymptotics for polygons. This could be used to measure the spectral determinant and Casimir energy of polygons whenever the first several hundred or one thousand Dirichlet eigenvalues are known with high precision by other means.

Subjects/Keywords: 515; Local spectral asymptotics; Heat kernel; Dirac operators; Laplace operators; Pseudo-differential operators; Fourier integral operators; Wodzicki residue; Finite propagation speed; Spectral determinant

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APA (6th Edition):

Li, L. (2016). Local spectral asymptotics and heat kernel bounds for Dirac and Laplace operators. (Doctoral Dissertation). Loughborough University. Retrieved from http://hdl.handle.net/2134/23004

Chicago Manual of Style (16th Edition):

Li, Liangpan. “Local spectral asymptotics and heat kernel bounds for Dirac and Laplace operators.” 2016. Doctoral Dissertation, Loughborough University. Accessed January 18, 2021. http://hdl.handle.net/2134/23004.

MLA Handbook (7th Edition):

Li, Liangpan. “Local spectral asymptotics and heat kernel bounds for Dirac and Laplace operators.” 2016. Web. 18 Jan 2021.

Vancouver:

Li L. Local spectral asymptotics and heat kernel bounds for Dirac and Laplace operators. [Internet] [Doctoral dissertation]. Loughborough University; 2016. [cited 2021 Jan 18]. Available from: http://hdl.handle.net/2134/23004.

Council of Science Editors:

Li L. Local spectral asymptotics and heat kernel bounds for Dirac and Laplace operators. [Doctoral Dissertation]. Loughborough University; 2016. Available from: http://hdl.handle.net/2134/23004

2. DEL CORRAL MARTINEZ, CESAR AUGUSTO. Canonical Trace and Pseudo-differential Operators on Manifolds with Boundary .

Degree: 2015, Universidad de los Andes

Esta tesis es acerca de la existencia y unicidad de la traza canónica para operadores pseudo-diferenciales de tipo log-polihomogeneo sobre una variedad con frontera. Por un lado, la traza canónica para operadores sobre una variedad cerrada (variedad compacta y sin frontera) fue definida, por Kontsevich, M. y Vishik, S., sobre una clase de operadores pseudo-diferenciales de orden no entero o de orden más pequeño que la dimensión de la variedad. Por otro lado, en el caso de una variedad con frontera L. Boutet De Monvel construye una álgebra de operadores pseudo-diferenciales asociado a problemas en la frontera, los operadores en tal álgebra requieren satisfacer la llamada propiedad de transmisión, tal propiedad requiere que el operador sea de orden entero, por lo cual el álgebra de Boutet de Monvel no es un espacio adecuado para estudiar la traza canónica. En esta tesis consideramos una clase apropiada de operadores pseudo-diferenciales de tipo log-polihomogeneo sobre una variedad con frontera, y extendemos la traza canónica a tal clase. Usando las técnicas empleadas por Fedosov et al., para estudiar el llamado residuo de Wodzicki o residuo no conmutativo para el caso de una variedad con frontera, nosotros probamos la tracialidad de la traza canónica (es decir, la traza canónica se anula sobre brackets de operadores pseudo-diferenciales). Al final de esta tesis, deducimos la unicidad de la traza canónica en el caso de variedades con frontera desde la unicidad de la traza canónica para variedades sin frontera. Advisors/Committee Members: Cardona Guio Alexander (advisor), Sylvie Paycha (advisor).

Subjects/Keywords: Fourier transform; Mellin transform; Paley-Wiener theorem; Transmission property; Pseudo-differential operator; Manifold with boundary; Truncated pseudo-differential operators; Wodzicki residue; Noncommutative residue; Canonical trace; Boundary traciality defect; Traciality

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

DEL CORRAL MARTINEZ, C. A. (2015). Canonical Trace and Pseudo-differential Operators on Manifolds with Boundary . (Thesis). Universidad de los Andes. Retrieved from https://documentodegrado.uniandes.edu.co/documentos/10575.pdf

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

DEL CORRAL MARTINEZ, CESAR AUGUSTO. “Canonical Trace and Pseudo-differential Operators on Manifolds with Boundary .” 2015. Thesis, Universidad de los Andes. Accessed January 18, 2021. https://documentodegrado.uniandes.edu.co/documentos/10575.pdf.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

DEL CORRAL MARTINEZ, CESAR AUGUSTO. “Canonical Trace and Pseudo-differential Operators on Manifolds with Boundary .” 2015. Web. 18 Jan 2021.

Vancouver:

DEL CORRAL MARTINEZ CA. Canonical Trace and Pseudo-differential Operators on Manifolds with Boundary . [Internet] [Thesis]. Universidad de los Andes; 2015. [cited 2021 Jan 18]. Available from: https://documentodegrado.uniandes.edu.co/documentos/10575.pdf.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

DEL CORRAL MARTINEZ CA. Canonical Trace and Pseudo-differential Operators on Manifolds with Boundary . [Thesis]. Universidad de los Andes; 2015. Available from: https://documentodegrado.uniandes.edu.co/documentos/10575.pdf

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

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