Weyl law and eigenvalue counting

For a closed d-dimensional smooth Riemannian manifold, the number of eigenvalues up to a large threshold is asymptotic to volume times the Euclidean unit-ball volume, divided by $(2\pi)^d$, times the threshold to the power d over two. The same leading law holds for bounded smooth Euclidean domains with standard Dirichlet or Neumann conditions. Frequency counting uses the square root of the eigenvalue and therefore has power d instead, a distinction that must be made before fitting an observed growth rate.

The constant arises by counting covectors whose metric squared length is below the threshold. Each spatial point contributes a frequency ball, and the phase-space cell normalization supplies the factors of two pi. In an explicit box, separation of variables turns this into counting lattice points in an ellipsoid. Boundary lattice points and the axes affect lower-order terms, while the ellipsoid volume gives the leading contribution. Bracketing transfers this local counting picture to curved or irregularly shaped regions in the standard theorem settings.

The exponent determines dimension, and after dimension is known the leading coefficient determines volume. In dimension two the coefficient is area divided by four pi. In dimension one it is length divided by pi when both directions of periodic frequency are counted. A finite number of zero modes changes only a bounded part of the count and cannot change either inference. Isospectral compact objects in the same scalar setting consequently have the same dimension and volume.

Inverting the leading asymptotic gives the growth of the kth eigenvalue, with the same multiplicity convention as the counting function. This is useful for estimating computational resolution, but it is not a certified bound for a small index without a separate remainder estimate. High-order numerical accuracy for the first few modes does not substitute for an asymptotic theorem, and conversely the leading asymptotic may be quite inaccurate at a threshold containing only a handful of modes.

Smooth boundary contributes a half-order term to the heat trace, but a matching two-term unsmoothed counting law is a more delicate assertion. It can require dynamical assumptions controlling periodic billiard trajectories. One should not obtain it by formally reading every heat coefficient as a pointwise counting coefficient. Tauberian arguments recover the leading law under broad hypotheses; sharper remainders and secondary terms demand more information about geometry and dynamics.

Corners, cusps, noncompact ends, and singular metrics also change the appropriate framework. A bounded rectangle still has a valid leading law despite its corners, but corner terms appear in finer heat expansions. On a noncompact manifold, discrete mode counts may not describe the full spectrum at all. Each extension needs an explicit operator and geometric class. The power of Weyl's law comes from a broad leading universality, not from erasing these distinctions in every more precise claim.