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Weyl asymptotics for non self adjoint operators with small random perturbations

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41 pages
Weyl asymptotics for non-self-adjoint operators with small random perturbations Johannes Sjostrand IMB, Universite de Bourgogne, UMR 5584 CNRS Resonances CIRM, 23/1, 2009 Johannes Sjostrand ( IMB, Universite de Bourgogne, UMR 5584 CNRS)Weyl asymptotics for non-self-adjoint operators with small random perturbationsResonances CIRM, 23/1, 2009 1 / 22

  • problems appear naturally

  • using standard

  • weyl asymptotics

  • self-adjoint operators

  • schrodinger operator


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JohannesI(dnU,BMo¨jSartsde´eBoveniitrsoticymptylasWejda-fles-nonrofsateropntoiseRnano192/2
IMB,Universit´edeBourgogne,UMR5584CNRS
Weyl asymptotics for non-self-adjoint operators with small random perturbations
JohannesSjo¨strand
Resonances CIRM, 23/1, 2009
cesCIRM,23/1,200
dortnI.1noitcuahnnoJo¨tssejSsamytptocifsroonWeyle´tioBedndraMB(Ini,UrsvesCIRanceesonRaterpotniojda-fles-n
yhxV0(x)hy2+γ(yhy)(y+hy).
A major difficulty is that the resolvent may be very large even when the spectral parameter is far from the spectrum:
1. Introduction
Non-self-adjoint spectral problems appear naturally e.g.: Resonances, (scattering poles) for self-adjoint operators, like the Schro¨dingeroperator, The Kramers–Fokker–Planck operator
σ(P) = spectrum ofP implies that. Thisσ(P) is unstable under small perturbations of the operator. (HereP:H → His a closed operator andHa complex Hilbert space.)
1 1 k(zP)k dist(z, σ(P)),
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1rodu.IntnctioWalyempsytitofocsonrnna(dMI,BnUvireist´edeBoneanohJtrosj¨sS
σ(P) = spectrum ofP implies that. Thisσ(P) is unstable under small perturbations of the operator. (HereP:H → His a closed operator andHa complex Hilbert space.)
A major difficulty is that the resolvent may be very large even when the spectral parameter is far from the spectrum:
yhxV0(x)hy2+γ(yhy)(y+hy).
Non-self-adjoint spectral problems appear naturally e.g.: Resonances, (scattering poles) for self-adjoint operators, like the Sch¨odingeroperator, r The Kramers–Fokker–Planck operator
k(zP)1k dist(z,1σ(P)),
2/09
1. Introduction
223/,2201,ICMRcnseosaneRtarpetoinjoadf-el-s
I.tn1tcoiorudn
using standard multiindex notation. Ifz=p(x, ξ),i1{p,p}(x, ξ)>0, thenu=uhC0(neigh(x,Rn)) such thatkukL2= 1, k(Pz)uk=O(h),h0. Butz Seemay be far from the spectrum! examples below.
be a differential operator with smooth coefficients on some open set inRn, with leading symbol p(x, ξ) =Xaα(x)ξα, |α|≤m
2
In the case of (pseudo)differential operators, this follows from the H¨ormander(1960)DaviesZworskiquasimode construction: Let
P=P(x,hDx) =Xaα(x)(hDx)α,Dx=i1x, |α|≤m
93/2,20023/1RI,MecCsnonaRsesalyeWBode´eBMU,dnI(srtiinevnnesJohastraSj¨oatesfla-jdiotnporeymptoticsfornon-