Spectral Zeta Function: Can You Hear the Shape?
Laplacian eigenvalues • Weyl's law • heat kernel trace • isospectral drums
Drum Geometry
Domain shape
Circle
Square
Rectangle
Right triangle
Isospectral drum A
Isospectral drum B
Mode number n
1
Animation speed
1.0
−Δu = λu (Helmholtz)
u|∂Ω = 0 (Dirichlet)
Spectral zeta:
ζ_Ω(s) = Σ λₙ⁻ˢ
Weyl's law:
N(λ) ~ |Ω|λ/4π
Kac (1966): can you hear
the shape of a drum?
λₙ ≈
–
f ∝ √λ =
–
Hz
Toggle Animation
Gordon, Webb & Wolpert (1992) answered Kac's question: NO — two different drum shapes can have identical eigenvalue spectra (isospectral). The spectral zeta function encodes area, perimeter, and Euler characteristic via its heat kernel expansion.