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 C^{2} boundary, from the spectrum
of an elliptic operator, typically the Laplacian
or Hamiltonian \(\square\) = *H*_{Σ} 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* = *H*_{Σ}

tr e^{−βH} = (2π *β*)^{−1} |*Σ*| − \(1\over4\)(2π *β*)^{−1/2} |∂*Σ*| + \(1\over6\)(1 − *m*) + *O*(*β*^{1/2}) ;

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].

@ __Non-commutative__: Martinetti a1502-proc [drum design for the truncated music of the spectral action]; > s.a. non-commutative theories.

@ __Other variations__: Kopf IJMPA(98)gq/96,
IJMPB(00)ht-in [Lorentzian]; D'Andrea et al JGP(14)-a1305 [with a cutoff].

**References**

@ __General__: Kac AMM(66); > 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].

@ __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__: P Martin, AIHP tbp(95) [& seminar Genève 15.12.1995];
Brezov a0805 [and semiclassical approach].

> __Online resources__: Wikipedia general page and Hearing-the shape-of-a-drum page.

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send feedback and suggestions to bombelli at olemiss.edu – modified 30
apr 2017