Spectral
Geometry| Spectral
Geometry |
In General [> s.a. laplacian.]
* Idea: The general question
is, Can one determine completely the geometry of a manifold, typically a bounded
portion 
R3 with
piecewise C2 boundary, from the spectrum
of an elliptic operator, typically the Laplacian
or Hamiltonian 
= HSigma for
a free particle with boundary condition
|bdry =
0, on it? Or, as M Kac put it, "Can one hear the shape of a drum?"
* History: G Gamow speculated
on what it would be like to play quantum billiard.
* Answer: Generically yes, but there are exceptions.
* Results: If m is the number of holes in ,
if there are corners, they also contribute terms (a few more terms are known).
* Remarks: Since tr(...) is essentially the propagator for a diffusion
process, we can interpret in terms of which aspects of the
particle
feels sooner.
Special Cases and Applications > s.a. approaches
to quantum gravity; dirac
fields in curved spacetime.
@ References: Arcos et al AJP(98)
[soap films and quantum chaos]; Gnutzmann et al PRL(06) [surfaces of revolution]; > s.a.
quantum chaos.
> In gravitation and cosmology:
see kerr-newman black holes; topology
of the universe.
Variations > s.a. graphs.
* From heat equation: The area,
circumference, and the number of holes in a planar domain can be recovered
from the short time
asymptotics of the solution of the initial-boundary value problem for the heat
equation.
* From wave equation: The
length spectrum of closed billiard ball trajectories in the domain can be recovered
from the eigenvalues or from the solution of the wave equation.
@ For Riemann tensor: Gilkey et al m.DG/02;
Stavrov T&A(07) [using vector bundles over Grassmannians].
@ For other operators: Blazic et al m.DG/03 [Weyl
tensor]; Schuss & Spivak mp/05 [from
trace of heat kernel].
@ Lorentzian: Kopf IJMPA(98)gq/96,
IJMPB(00)ht-in.
References
@ Isospectral manifolds: Szabó AM(01) [spheres]; > s.a. 3D
geometry.
@ Experiment: Gordon & Webb AS(96)
[2 drums with same f 's];
Wilkinson et
al Nat(96) + pn(96)may
[2D
quantum chaos].
@ With Dirac operator: Martinetti m.QA/07 [spectral
distance on the circle];
>
s.a. dirac
fields
in curved spacetime.
@ Related topics: P Martin, AIHP tbp(95) [& seminar Genève
15.12.1995];
Brezov a0805 [and semiclassical approach].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
23 may 2008