Localization of the LaplacianLaplacianEigenfunctions ... · (by Chris King, (by Chris King, ... irregularirregularor fractal drums by B. or fractal drums by B. SapovalSapoval et al.

Localization of the Localization of the

LaplacianLaplacian EigenfunctionsEigenfunctions in in

IrregularlyIrregularly--Shaped DomainsShaped DomainsIrregularlyIrregularly--Shaped DomainsShaped Domains

A PhD thesis of A PhD thesis of BinhBinh ThanhThanh NguyenNguyen

44thth CODY Conference, 2CODY Conference, 2--8 May 2010, 8 May 2010, SeillacSeillac, France, France

Denis Denis GrebenkovGrebenkovLaboratory of Condensed Matter Physics,Laboratory of Condensed Matter Physics,

CNRS CNRS –– EcoleEcole PolytechniquePolytechnique FranceFrance

Outline of the talkOutline of the talk

• Introduction and short • Introduction and short overviewoverview

• • LocalizationLocalization and and conformalconformal mappingmapping

• Open questions and • Open questions and furtherfurther perspectivesperspectives• Open questions and • Open questions and furtherfurther perspectivesperspectives

What is an What is an eigenfunctioneigenfunction??

AcousticsAcoustics: : vibration modes vibration modes

Quantum Quantum mechanicsmechanics:: wavefunctionswavefunctions

LinearLinear algebraalgebra: : eigenvectorseigenvectors

Diffusion: Diffusion: eigenfunctionseigenfunctionsDiffusion: Diffusion: eigenfunctionseigenfunctions

a a linearlinear operatoroperator//matrixmatrix

an an eigenvalueeigenvaluean an eigenfunctioneigenfunction

In In acousticsacoustics: : vibration modes vibration modes

What is an What is an eigenfunctioneigenfunction??

((fromfrom the the webpagewebpage by L. by L. SaviotSaviot) )

In quantum In quantum mechanicsmechanics: :

wavefunctionswavefunctions or or eigenstateseigenstates

What is an What is an eigenfunctioneigenfunction??

(A (A wavefunctionwavefunction of an of an electronelectron in a in a hydrogenhydrogen atomatom, by B. , by B. CrowellCrowell))

For diffusion:For diffusion: LaplacianLaplacian eigenfunctionseigenfunctions

What is an What is an eigenfunctioneigenfunction??

What do we know and What do we know and

what do we want to know?what do we want to know?

General spectral General spectral propertiesproperties

EigenfunctionsEigenfunctions for simple for simple shapesshapes

WeWe know:know:

AsymptoticAsymptotic behaviorbehavior of of eigenvalueseigenvalues

Relation Relation eigenfunctionseigenfunctions//geometrygeometry??

LocalizationLocalization of of eigenfunctionseigenfunctions

PhysicalPhysical consequencesconsequences and applicationsand applications

WeWe wantwant to know:to know:

Quantum billiardsQuantum billiards

(by Chris King, (by Chris King, UniversityUniversity of Auckland)of Auckland)

ScarringScarring

Quantum billiardsQuantum billiards

G. G. VebleVeble et et alal,New,New J. Phys. 9, 15 (2007)J. Phys. 9, 15 (2007)

EigenfunctionsEigenfunctions on fractalson fractals

M. M. LapidusLapidus et al. ,Int. J. et al. ,Int. J. BifurBifur. Chaos 6, 1185 (1996). Chaos 6, 1185 (1996)

The 13The 13thth eigenfunctioneigenfunction on a von Koch snowflakeon a von Koch snowflake

Its gradientIts gradient

Nodal linesNodal lines

EigenfunctionsEigenfunctions on fractalson fractals

M. M. LapidusLapidus et al. ,Int. J. et al. ,Int. J. BifurBifur. Chaos 6, 1185 (1996). Chaos 6, 1185 (1996)

What is done at PMC?What is done at PMC?

SinceSince 1990s, 1990s, manymany studiesstudies of vibrations of of vibrations of

irregularirregular or fractal drums by B. or fractal drums by B. SapovalSapoval et al.et al.

(B. (B. SapovalSapoval, T. Gobron, A. , T. Gobron, A. MargolinaMargolina, Phys. , Phys. RevRev. . LettLett. 67, 2974, 1991). 67, 2974, 1991)

What is done at PMC?What is done at PMC?

SinceSince 1990s, 1990s, manymany studiesstudies of vibrations of of vibrations of

irregularirregular or fractal drums by B. or fractal drums by B. SapovalSapoval et al.et al.

(S. (S. Russ, B. Russ, B. SapovalSapoval, Phys. Rev. E 65, 036614, 2002), Phys. Rev. E 65, 036614, 2002)

What is done at PMC?What is done at PMC?

SinceSince 1990s, 1990s, manymany studiesstudies of vibrations of of vibrations of

irregularirregular or fractal drums by B. or fractal drums by B. SapovalSapoval et al.et al.

(S. (S. FelixFelix, M. Asch, M. Filoche, B. , M. Asch, M. Filoche, B. SapovalSapoval, J. Sound. , J. Sound. VibrVibr. 299, 965, 2007. 299, 965, 2007))

What is done at PMC?What is done at PMC?

SinceSince 1990s, 1990s, manymany studiesstudies of vibrations of of vibrations of

irregularirregular or fractal drums by B. or fractal drums by B. SapovalSapoval et al.et al.

… … towardstowards one of (one of (manymany) ) practicalpractical applicationsapplications

(Anti(Anti--noise noise wallswalls))

What is done at PMC?What is done at PMC?

SinceSince 1990s, 1990s, manymany studiesstudies of vibrations of of vibrations of

irregularirregular or fractal drums by B. or fractal drums by B. SapovalSapoval et al.et al.

(Rough) (Rough) summarysummary: : (Rough) (Rough) summarysummary: :

geometricalgeometrical irregularityirregularity maymay leadlead

to to localizationlocalization of of eigenfunctionseigenfunctions

Localization of Localization of eigenfunctionseigenfunctions

”Regular” domains”Regular” domains(interval, disk, sphere)(interval, disk, sphere)

Nonlocalized (extended) eigenfunction:Nonlocalized (extended) eigenfunction:Eigenfunction is supported by the whole Eigenfunction is supported by the whole domaindomain

Localized eigenfunction:Localized eigenfunction:There exists a small compact subset that There exists a small compact subset that supports a large fraction of eigenfunctionsupports a large fraction of eigenfunction ??

Localization of Localization of eigenfunctionseigenfunctions

Localization of Localization of eigenfunctionseigenfunctions

Geometrical irregularity Geometrical irregularity

Localized eigenfunctionsLocalized eigenfunctions

Localization of Localization of eigenfunctionseigenfunctions

Trapping/exchange processTrapping/exchange process

AAParticular structure of BParticular structure of B

What is the role of localization What is the role of localization in diffusive processes?in diffusive processes?

“Geometry“Geometry--induced field”induced field”

conformalconformal mapmap

“Geometry“Geometry--induced field”induced field”

conformalconformal mapmap

D’D’

A’A’

“Geometry“Geometry--induced field”induced field”

ExampleExample

AA BB

CCDD

B’B’

C’C’

D’D’

A’A’

“Geometry“Geometry--induced field”induced field”

ExampleExample

AA BB

CCDD

B’B’

C’C’

D’D’

A’A’

“Geometry“Geometry--induced field”induced field”

ExampleExample

AA BB

CCDD

B’B’

C’C’

D’D’

A’A’

“Geometry“Geometry--induced field”induced field”

ExampleExample

AA BB

CCDD

B’B’C’C’

D’D’

A’A’

“Geometry“Geometry--induced field”induced field”

ExampleExample

AA BB

CCDD

B’B’C’C’

Screening and localizationScreening and localization

ExampleExample

Screening and localizationScreening and localization

ExampleExample

“Conformal microscope”“Conformal microscope”

Spectral approachSpectral approach

Complex Complex geometrygeometry

Disk + Disk + conformal mapconformal map

Diffusion inDiffusion in Diffusion inDiffusion inDiffusion inDiffusion incomplex geometrycomplex geometry

Diffusion inDiffusion incomplex potentialcomplex potential

Localization byLocalization bycomplex geometrycomplex geometry

Localization byLocalization bycomplex potentialcomplex potential

Anderson localization?Anderson localization?

Questions to answerQuestions to answer

Geometry Geometry -- conformal map conformal map -- matrix B?matrix B?

Why and when does localization emerge?Why and when does localization emerge?

What are the properties of eigenfunctions What are the properties of eigenfunctions in complex geometries?in complex geometries?in complex geometries?in complex geometries?

What is the role of localization for diffusive What is the role of localization for diffusive processes?processes?

What are the physical consequences of What are the physical consequences of localization, e.g., in NMR experiments?localization, e.g., in NMR experiments?