Weyl asymptotics for non self adjoint operators with small random perturbations

Johannes Sjostrand

Description

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

Sujets

• Sjöstrand
• Self-adjoint operator
• Schrödinger equation

Informations

• Publié par: pefav
• Nombre de lectures: 12
• Langue: English

Extrait

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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

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y∙h∂ x −V 0 (x)∙h∂ y 2 + 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)) ,

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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 −V 0 (x)∙h∂ y 2 + 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) −1 k  dist(z, 1 σ(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=u h ∈C 0 ∞ (neigh(x,R n )) such thatkuk L 2 = 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 inR n , with leading symbol p(x, ξ) = X a α (x)ξ α , |α|≤m

2

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

P=P(x,hD x ) = X a α (x)(hD x ) α ,D x = i 1 ∂ ∂ x , |α|≤m

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