SPECTRAL ASYMPTOTICS FOR MAGNETIC LAPLACIANS IN HYPERBOLIC GEOMETRY

Description

SPECTRAL ASYMPTOTICS FOR MAGNETIC LAPLACIANS IN HYPERBOLIC GEOMETRY Franc¸oise Truc Institut Fourier, Grenoble 16/11/2009 – p. 1

• geometrically finite hyperbolic


• riemannian manifold


• b˜ dv


• hyperbolic geometry


• surfaces magnetic bottles


• magnetic bottles


• iff b˜

Sujets

• Riemannian manifold
• Hyperbolic geometry

Informations

• Publié par: pefav
• Publié le: 01 novembre 2009
• Nombre de lectures: 5
• Langue: English

Extrait

SPECTRAL ASYMPTOTICS FOR MAGNETIC

LAPLACIANS IN HYPERBOLIC GEOMETRY

Franc¸ oise Truc

Institut Fourier, Grenoble

161/1/2009–p.1

•

•

•

The hyperbolic context

The Poincaré Half-plane

Outline

The constant magnetic Laplacian

Magnetic bottles : compacity of the resolvent

Magnetic bottles : spectral asymptotics

Sketch of the proof

Related results

Geometrically nite hyperbolic surfaces

Magnetic bottles : spectral asymptotics

The spectrum of constant magnetic Laplacians

The Weyl formula in the case of nite area with a non-integer class one-form

Oultineofproofs 16/11/2009–p.2

Framework

Let(gM)be a connected and oriented Riemannian manifold of dimensionn.

For any real one-formAonM dene

−Δ A = (i d+A) ⋆ (i d+A)((i d+A)u=i du+uA ∀u

The magnetic eld is the two-formdA.

1

∈

61/1/ C 20 ∞ 0 09 ( – M .p3 ))

M)) A Δ(=diA+−(i(A)d+(i) ⋆ A)d+∈u∀( 0 ∞ Cdi=uAu+u 2/11/613p.9–00

The magnetic intensitybis given by b= 1 2 tr  (B ⋆ B) 12  

TodAis associated the linear operatorBdened on the tangent space by

dA(X Y) =g(BX Y) ;∀X  Y∈T M× T M

For any real one-formAonM dene

The magnetic eld is the two-formdA.

Let(gM)be a connected and oriented Riemannian manifold of dimensionn.

Framework

The Poincaré half-plane

e Ifdim(M) = 2thendA=bdv with

dvthe Riemannian measure onM.

The

magnetic

eld

is

constant

iff

e b

is

e |b|=b

constant.



16/11/2009

–

p.

4

The Poincaré half-plane

e e Ifdim(M) = 2thendA=bdv with|b|=b dvthe Riemannian measure onM. e The magnetic eld is constant iffbis constant.

LetM=Hbe the hyperbolic half-plane : H=R×]0+ [ g= dx 2 +dy 2  ∞ y 2 −Δ A =y 2 (D x −A 1 ) 2 +y 2 (D y −A 2 ) 2  withA=A 1 (x y)dx+A 2 (x y)dy and A j (x y)∈C ∞ (H;R) e b=y 2 (∂ x A 2 −∂ y A 1 ) e b=|b| dv=y −2 dxdy 

16/112/009–p.4