Weyl asymptotics for non self adjoint operators with small random perturbations

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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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pefav

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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

y∙h∂x−V0(x)∙h∂y2+γ(y−h∂y)∙(y+h∂y).

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(z−P)k dist(z, σ(P)),

290022/32,M2,1/

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:

y∙h∂x−V0(x)∙h∂y2+γ(y−h∂y)∙(y+h∂y).

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(z−P)−1k dist(z,1σ(P)),

2/09

1. Introduction

223/,2201,ICMRcnseosaneRtarpetoinjoadf-el-s

I.tn1tcoiorudn

using standard multiindex notation. Ifz=p(x, ξ),i−1{p,p}(x, ξ)>0, then∃u=uh∈C0∞(neigh(x,Rn)) such thatkukL2= 1, k(P−z)uk=O(h∞),h→0. 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

In the case of (pseudo)diﬀerential operators, this follows from the H¨ormander(1960)–Davies–Zworskiquasimode construction: Let

P=P(x,hDx) =Xaα(x)(hDx)α,Dx=i1∂∂x, |α|≤m

93/2,20023/1RI,MecCsnonaRsesalyeWBode´eBMU,dnI(srtiinevnnesJohastraSj¨oatesfla-jdiotnporeymptoticsfornon-

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