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].

@ __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