Questo sito utilizza cookie di terze parti per inviarti pubblicità in linea con le tue preferenze. Se vuoi saperne di più clicca QUI 
Chiudendo questo banner, scorrendo questa pagina, cliccando su un link o proseguendo la navigazione in altra maniera, acconsenti all'uso dei cookie. OK

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.