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 Σ ⊂ \(\mathbb R\)3
with piecewise C2 boundary, from the spectrum
of an elliptic operator, typically the Laplacian or Hamiltonian \(\square\)
= HΣ for a free particle
with boundary condition ψ|bdry = 0,
on it? Or, as Mark Kac put it in 1966, "Can one hear the shape of a drum?"
* History: G Gamow speculated on
what it would be like to play quantum billiard.
* Answer: Carolyn Gordon and
colleagues in 1992 said no, in general; Generically yes, but there are exceptions.
* Results: If m is the
number of holes in the spatial region Σ, then for the Hamiltonian
H = HΣ
tr e−βH = (2π β)−1 |Σ| − \(1\over4\)(2π β)−1/2 |∂Σ| + \(1\over6\)(1 − m) + O(β1/2) ;
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 the expression
in terms of which aspects of Σ the particle feels sooner.
Special Cases and Applications
@ References: Arcos et al AJP(98)jul [soap films and quantum chaos];
Gnutzmann et al PRL(06) [surfaces of revolution];
> s.a. quantum chaos.
@ In gravitation and cosmology:
Panine & Kempf PRD(16)-a1601 [linearized spectral geometry and euclidean quantum gravity];
> s.a. kerr-newman black holes;
topology of the universe.
> Quantum-gravity related:
see modified approaches; hořava gravity;
causal set kinematics.
> Other theories: see dirac
fields in curved spacetime; types of spinors [symplectic spinors].
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;
Yazdi et al a2008 [with causal sets].
@ Non-commutative: Martinetti a1502-proc [drum design for the truncated music of the spectral action];
> s.a. non-commutative theories.
@ Other variations: D'Andrea et al JGP(14)-a1305 [with a cutoff].
References
@ General: Kac AMM(66);
Urakawa 17;
> s.a. 3D geometry.
@ Isospectrality: Szabó AM(01) [on spheres];
Giraud & Thas RMP(10)-a1101;
Amore PRE(13)-a1307;
Liu et al a1701 [3D case];
Thas a1712 [D-geometry].
@ Experiment:
Gordon & Webb BAMS(92),
AS(96) [two drums with the same frequencies];
Wilkinson et al Nat(96)
+ pn(96)may [2D quantum chaos].
@ With Dirac operator: Martinetti JFA(08)m.QA/07 [spectral distance on the circle];
Wallet RVMP(12)-a1112 [examples];
> s.a. dirac fields in curved spacetime.
@ Related topics: Martin et al AIHP(97) [& seminar Genève 1995-12-15];
Brezov a0805 [and semiclassical approach]; Lu & Rowlett BLMS(16)-a2012 [one can hear corners].
> Online resources:
Wikipedia general page and
Hearing-the shape-of-a-drum page.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 14 dec 2020