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].
@ 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); 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 15.12.1995]; Brezov a0805 [and semiclassical approach].
> Online resources: Wikipedia general page and Hearing-the shape-of-a-drum page.