Eigenmodes of planar drumsand Physics Nobel PrizesNick TrefethenNumerical Analysis GroupOxford University Computing Laboratory

Consider an idealized drum:Laplace operator in 2D regionwith zero boundary conditions.Basis of the presentation: numerical methods developed with Timo Betcke, U. of ManchesterB. & T., Reviving the method of particular solutions, SIAM Review, 2005

T. & B., Computed eigenmodes of planar regions, Contemporary Mathematics, 2006

Contributors to this subject includeBarnettCohenHellerLeporeVerginiSaraceno

BanjaiBetckeDesclouxDriscollKarageorghisTolleyTrefethen

ClebschPockelsPoissonRayleighLamSchwarzWeber

GoldstoneJaffeLenzRavenhallSchultWyldDonnellyEisenstatFoxHenriciMasonMolerShryerBenguriaExnerKuttlerLevitinSigillitoBerryGutzwillerKeatingSimonWilkinsonBckerBoasmannRiddelSmilanskySteinerConwayFarnhamand many more.

EIGHT EXAMPLES 1. L-shaped region 2. Higher eigenmodes 3. Isospectral drums 4. Line splitting 5. Localization 6. Resonance 7. Bound states 8. Eigenvalue avoidance

[Numerics]Eigenmodes of drums

Physics Nobel Prizes

PLAN OF THE TALK

1. L-shaped regionFox-Henrici-Moler 1967

Cleve Moler, late 1970sMathematically correct alternative:

Some higher modes, to 8 digitstriple degeneracy: 52 + 52 =12 + 72 =72 + 12

Circular L shapeno degeneracies, so far as I know

Schrdinger 1933Schrdingers equation and quantum mechanicsalso Heisenberg 1932, Dirac 1933

2. Higher eigenmodes

Eigenmodes 1-6

Eigenmodes 7-12

Eigenmodes 13-18

Six consecutive higher modes

Weyls Lawn ~ 4n / A( A = area of region )

Planck 1918blackbody radiationalso Wien 1911

3. Isospectral drums

Kac 1966Can one hear the shape of a drum? Gordon, Webb & Wolpert 1992Isospectral plane domains and surfaces via Riemannian orbifolds

Numerical computations:Driscoll 1997Eigenmodes of isospectral drums

Microwave experiments Sridhar & Kudrolli 1994 Computations with DTD method Driscoll 1997 Holographic interferometry Mark Zarins 2004

A MATHEMATICAL/NUMERICAL ASIDE: WHICH CORNERS CAUSE SINGULARITIES?That is, at which corners are eigenfunctions not analytic? (Effective numerical methods need to know this.)Answer: all whose angles are / , integerProof: repeated analytic continuation by reflection

E.G., the corners marked in red are the singular ones:

Michelson 1907spectroscopyalso Bohr 1922, Bloembergen 1981

4. Line splitting

Bongo drumstwo chambers, weakly connected.Without the coupling the eigenvalues are { j 2 + k 2 } = { 2, 2, 5, 5, 5, 5, 8, 8, 10, 10, 10, 10, }

With the coupling

Now halve the width of the connector.( = 1.3%)( = 0.017%)

( = 1.3%)( = 4.1%)( = 0.005%)

Zeeman 1902Zeeman & Stark effects: line splitting in magnetic & electric fieldsalso Michelson 1907, Dirac 1933, Lamb & Kusch 1955Stark 1919

5. Localization

What if we make the bongo drums a little asymmetrical? + 0.2 + 0.2

GASKET WITH FOURFOLD SYMMETRYNow we will shrink this hole a little bit

GASKET WITH BROKEN SYMMETRY

Anderson 1977Anderson localizationand disordered materials

6. Resonance

A square cavity coupled to a semi-infinite strip of width 1slightly above the minimum frequency 2 The spectrum iscontinuous: [2,)Close to the resonant frequency 22

Lengthening the slit strengthens the resonance:

Marconi 1909also Bloch 1952Purcell 1952RadioNuclear Magnetic Resonance

Townes 1964Masers and lasersalso Basov and Prokhorov 1964Schawlow 1981

7. Bound states

See papers by Exner and 1999 book by Londergan, Carini & Murdock.

Marie Curie 1903Radioactivityalso Becquerel and Pierre Curie 1903 Rutherford 1908 [Chemistry]Marie Curie 1911 [Chemistry]Irne Curie & Joliot 1935 [Chemistry]

8. Eigenvalue avoidance

Another example of eigenvalue avoidance

Wigner 1963Energy states of heavy nuclei eigenvalues of random matrices

cf. Montgomery, Odlyzko, The 1020th zero of the Riemann zeta function and 70 million of its neighbors Riemann Hypothesis?

Weve mentioned:Zeeman 1902Becquerel, Curie & Curie 1903Michelson 1907Marconi & Braun 1909Wien 1911Planck 1918Stark 1919Bohr 1922Heisenberg 1932Dirac 1933Schrdinger 1933Bloch & Purcell 1952Lamb & Kusch 1955Wigner 1963Townes, Basov & Prokhorov 1964Anderson 1977Bloembergen & Schawlow 1981We might have added:Barkla 1917Compton 1927Raman 1930Stern 1943Pauli 1945Mssbauer 1961Landau 1962Kastler 1966Alfvn 1970Bardeen, Cooper & Schrieffer 1972Cronin & Fitch 1980Siegbahn 1981von Klitzing 1985Ramsay 1989Brockhouse 1994Laughlin, Strmer & Tsui 1998

Thank you!