Spectral asymptotics of compact pseudodifferential operators with ?constant? symbol

AbstractThis is the third in a series of works devoted to spectral asymptotics for non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2, having a periodic classical flow. Assuming that the strength ॉ of the perturbation is in the range h2 ≪ ॉ ≪ h1/2 (and may sometimes reach even smaller values), we get an asymptotic description of the eigenvalues in rectangles [−1/C, 1/C] + iॉ[F0 − 1/C, F0 + 1/C], C ≫ 1, when ॉF0 is a saddle point value of the flow average of the leading perturbation.

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Improved spectral asymptotics and periodic geodesics

10.23943/princeton/9780691160757.003.0005 ◽

2017 ◽

Author(s):

Christopher D. Sogge

Keyword(s):

Geodesic Flow ◽

Measure Zero ◽

Pseudodifferential Operators ◽

Spectral Asymptotics ◽

Weyl Formula ◽

Norm Estimates ◽

Spectral Bands

This chapter proves an improved Weyl formula under the assumption that the set of periodic geodesics for (M,g) has measure zero. It then shows trace estimates associated with shrinking spectral bands, details and proves a lemma, and gives a related generalization of the Weyl formula from Chapter 3 that involves pseudodifferential operators. The chapter then proves its main result by using a version of the Duistermaat-Guillemin theorem, which allows the use of the Hadamard parametrix and the arguments from Chapter 3. To conclude, the chapter shows that one can improve the sup-norm estimates from Chapter 3 if one assumes a condition on the geodesic flow that is similar to a hypothesis laid out in the Duistermaat-Guillemin theorem.

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Two remarks about spectral asymptotics of pseudodifferential operators

Colloquium Mathematicum ◽

10.4064/cm-80-1-131-145 ◽

1999 ◽

Vol 80 (1) ◽

pp. 131-145

Author(s):

Wojciech Czaja ◽

Ziemowit Rzeszotnik

Keyword(s):

Pseudodifferential Operators ◽

Spectral Asymptotics

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Equivariant spectral asymptotics forh-pseudodifferential operators

Journal of Mathematical Physics ◽

10.1063/1.4896698 ◽

2014 ◽

Vol 55 (10) ◽

pp. 101501 ◽

Cited By ~ 1

Author(s):

Tobias Weich

Keyword(s):

Pseudodifferential Operators ◽

Spectral Asymptotics

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On the Spectral Asymptotics of Operators on Manifolds with Ends

Abstract and Applied Analysis ◽

10.1155/2013/909782 ◽

2013 ◽

Vol 2013 ◽

pp. 1-21 ◽

Cited By ~ 7

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.

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