Localization of the LaplacianLaplacianEigenfunctions ... · (by Chris King, (by Chris King, ......
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Localization of the Localization of the Laplacian Laplacian Eigenfunctions Eigenfunctions in in Irregularly Irregularly-Shaped Domains Shaped Domains A PhD thesis of A PhD thesis of Binh Binh Thanh Thanh Nguyen Nguyen 4 th th CODY Conference, 2 CODY Conference, 2-8 May 2010, 8 May 2010, Seillac Seillac, France , France Denis Denis Grebenkov Grebenkov Laboratory of Condensed Matter Physics, Laboratory of Condensed Matter Physics, CNRS CNRS – Ecole Ecole Polytechnique Polytechnique France France
Transcript of Localization of the LaplacianLaplacianEigenfunctions ... · (by Chris King, (by Chris King, ......
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?
