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Microlocal Analysis and Spectral Theory of Elliptic Operators on Non-compact Manifolds

In this thesis we will deal with SG-pseudodifferential operators, a class of global pseudodifferential operators introduced independently by H. O. Cordes and C. Parenti in the '70s. A main feature of this pseudodifferential operators class is the possibility to transfer the calculus to a rather wide set of non-compact manifolds, the so-called S-manifolds.
In Chapter 1 we will recall the basic definitions and results about the SG-calculus on R^n. Here we will mainly focus on three aspects: the problem of the existence of a parametrix for elliptic operators in this class, some properties of the weighted Sobolev spaces, and the analysis of a version of the calculus on the polycylinder.
The description of the S-structures will be given in Chapter 2. Here we will describe how the SG-calculus can be transferred to manifolds which admit a S-structure and show that the good properties of the calculus hold also on S-manifolds. We will also discuss the Laplace-Beltrami operator on such manifolds, and show that it is not md-elliptic.
In Chapter 3 we will then fix the attention to the spectral properties of elliptic SG-operators of positive order on S-manifolds. Finally, in some concluding remarks, we briefly discuss how the theory could be used to build explicit examples of estimates for the corresponding spectral counting functions.

Mostra/Nascondi contenuto.
Notations Throughout the text we utilize the following notations. x = (x 1 ;:::;x n ) will denote a point inR n . If x;y2R n , then x y =x 1 y 1 + +x n y n jxj = (x x) 1=2 = (x 2 1 + +x 2 n ) 1=2 hxi = p 1 +jxj 2 dx =dx 1 dx n d x = (2 ) n dx = (2 ) n dx 1 dx n GivenN =Z + =f0; 1; 2;:::g, we dene a multi-index as = ( 1 ;:::; n )2Z n + : If and are both multi-indices, then j j = 1 + + n ; , for all i; 1 i n : i i = ( 1 1 ; ; n n ); ( in ); = 1 1 n n ; ( ); @ = @ @x 1 1 @ @x n n [email protected] 1 1 @ n n : If x2R n and 2Z n + , then x =x 1 1 x n n ; where, if j = 0, we set x j j = 1. We dene the operator D , with 2Z n + , as : D =i j j @ = 1 i @ @x 1 1 1 i @ @x n n = ( [email protected] 1 ) 1 ( [email protected] n ) n : IfP is a polynomial of degreem2Z + inn variables with complex coecients c , P ( ) = X j j m c = X j j m c 1 1 n n ; ii

Laurea liv.II (specialistica)

Facoltà: Scienze Matematiche, Fisiche e Naturali

Autore: Massimo Borsero Contatta »

Composta da 57 pagine.


Questa tesi ha raggiunto 65 click dal 19/04/2011.

Disponibile in PDF, la consultazione è esclusivamente in formato digitale.