scholarly journals On the Spectral Asymptotics of Operators on Manifolds with Ends

2013 ◽  
Vol 2013 ◽  
pp. 1-21 ◽  
Author(s):  
Sandro Coriasco ◽  
Lidia Maniccia
Keyword(s):  
Asymptotic Behaviour ◽  
Pseudodifferential Operators ◽  
Counting Function ◽  
Integral Operators ◽  
Fourier Integral ◽  
Spectral Asymptotics ◽  
Fourier Integral Operators ◽  
Noncompact Manifolds ◽  
Adjoint Operators

We deal with the asymptotic behaviour, forλ→+∞, of the counting functionNP(λ)of certain positive self-adjoint operatorsPwith double order(m,μ),m,μ>0,  m≠μ, defined on a manifold with endsM. The structure of this class of noncompact manifolds allows to make use of calculi of pseudodifferential operators and Fourier integral operators associated with weighted symbols globally defined onℝn. By means of these tools, we improve known results concerning the remainder terms of the Weyl Formulae forNP(λ)and show how their behaviour depends on the ratiom/μand the dimension ofM.

scholarly journals On the geometry of $Diff(S^1)$-pseudodifferential operators based on renormalized traces.

2021 ◽  
Vol 14 (1) ◽  
pp. 19-47
Author(s):  
Jean-Pierre Magnot
Keyword(s):  
General Linear Group ◽  
Differential Operators ◽  
Pseudodifferential Operators ◽  
Integral Operators ◽  
Fourier Integral ◽  
Riemannian Metric ◽  
Fourier Integral Operators ◽  
Pseudo Differential Operators ◽  
Relationship Of ◽  
The Relationship

In this article, we examine the geometry of a group of Fourier-integral operators, which is the central extension of $Diff(S^1)$ with a group of classical pseudo-differential operators of any order. Several subgroups are considered, and the corresponding groups with formal pseudodifferential operators are defined. We investigate the relationship of this group with the restricted general linear group $GL_{res}$, we define a right-invariant pseudo-Riemannian metric on it that extends the Hilbert-Schmidt Riemannian metric by the use of renormalized traces of pseudo-differential operators, and we describe classes of remarkable connections.

scholarly journals $L^{p}$ Estimates for Joint Quasimodes of Semiclassical Pseudodifferential Operators Whose Characteristic Sets Have $k$th Order Contact

2020 ◽  
Author(s):  
Melissa Tacy
Keyword(s):  
Wavelet Analysis ◽  
Level Sets ◽  
Pseudodifferential Operators ◽  
Integral Operators ◽  
Fourier Integral ◽  
Technical Development ◽  
Fourier Integral Operators ◽  
Characteristic Sets ◽  
Semiclassical Pseudodifferential Operators ◽  
Order Contact

Abstract In this paper we develop $L^{p}$ estimates for functions $u$, which are joint quasimodes of semiclassical pseudodifferential operators $p_{1}(x,hD)$ and $p_{2}(x,hD)$ whose characteristic sets meet with $k$th order contact, $k\geq{}1$. As part of the technical development we use Fourier integral operators to adapt a flat wavelet analysis to the curved level sets of $p_{1}(x,\xi )$.

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