Weyl's asymptotic law
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You cannot hear the exact shape of a drum, but you can hear its SIZE. Pile up the overtones of a vibrating membrane and count how many lie below a given pitch; that count grows in a way that is dictated, in the long run, by nothing more than the area of the drum (in higher dimensions, the volume). Weyl's asymptotic law is the precise statement of this: the high-frequency eigenvalues of the Dirichlet Laplacian know the volume of the region, and the dimension, and almost nothing else.
Let N(lambda) be the number of Dirichlet eigenvalues less than or equal to lambda (counted with multiplicity). Weyl's law says that as lambda tends to infinity, N(lambda) is asymptotic to (omega_n / (2 pi)^n) times |Omega| times lambda^(n/2), where n is the dimension, |Omega| is the volume of the domain, and omega_n is the volume of the unit ball in n dimensions. The intuition is a phase-space count: each eigenmode occupies a fixed cell of volume (2 pi)^n in position-momentum space, and you are simply measuring how many such cells fit inside the region of phase space where the energy is below lambda. Inverting, the n-th eigenvalue itself grows like lambda_n is asymptotic to a constant times (n / |Omega|)^(2/n) — eigenvalues climb at a rate set purely by how big the domain is and how many dimensions it has.
This is the great two-way street between the spectrum and geometry. It says the leading-order spectrum determines the volume — so you CAN hear the area of a drum, even though Kac showed you cannot hear its full shape. Sharper expansions recover more geometry: the next term in N(lambda) involves the surface area of the boundary (with a sign depending on Dirichlet versus Neumann), and finer terms feel the curvature, a story that flowers into spectral geometry and the heat-kernel trace expansion. Weyl's law also underwrites physics — it is exactly the counting that gives the Rayleigh-Jeans density of electromagnetic modes in a cavity and the density of states in a quantum box.
On a 2D rectangle the eigenvalues are pi^2 (p^2/a^2 + q^2/b^2) for positive integers p, q. Counting lattice points (p, q) inside an ellipse of area proportional to lambda gives N(lambda) is asymptotic to (area of rectangle / (4 pi)) times lambda — exactly Weyl's law in dimension two, with the rectangle's area appearing as the coefficient. The number of overtones below pitch lambda grows linearly in lambda, at a rate fixed by the area alone.
Counting eigenvalues below lambda is, to leading order, just measuring the domain's volume.
Weyl's law is an ASYMPTOTIC statement about high eigenvalues — it says nothing reliable about the lowest few, which depend heavily on shape. And it shows the leading spectral term sees only volume, which is exactly why distinct shapes can share a spectrum (Kac's question): same area means same Weyl asymptotics, so the leading term cannot distinguish them.