Weyl's Law: Spectral Properties of the Laplacian in Mathematics and Physics

Abstract

Weyl’s law is in its simplest version a statement on the asymptotic growth of the eigenvalues of the Laplacian on bounded domains with Dirichlet and Neumann boundary conditions. In the typical applications in physics one deals either with the Helmholtz wave equation describing the vibrations of a string, a membrane (drum), a mass of air in a concert hall, the heat radiation from a body in thermal equilibrium, the fluctuations of the gravitational field in cosmology, or the Schrodinger equation of a quantum system which may be a simple quantum billiard, an atom, a molecule or a compound nucleus. While Weyl’s seminal work was provoked by the famous black body radiation problem, i.e. an electromagnetic cavity problem, in particular by a conjecture put forward independently by Sommerfeld and Lorentz in 1910, Weyl’s law has its roots in music and, respectively, acoustics. Already in 1877, Lord Rayleigh had, in his famous book, “The Theory of Sound” treated the overtones of a violin or piano string and the natural notes of an organ pipe or the air contained within a room. For a room of cubical shape he derived the correct asymptotic behavior for the overtones. The trick used by Rayleigh to count the vibrational modes was to reduce the problem to a three-dimensional lattice-point problem from which he could derive that the number of overtones with frequency between ν and ν + dν grows at high frequencies, ν → ∞, asymptotically as V ·ν3 (Weyl’s law!), where V is the volume of the room or analogously of an organ pipe. In 1900, Rayleigh realized that the same formula can be applied to a physically completely different, but mathematically equivalent problem: the heat radiation from a body in thermal equilibrium with radiation, the importance of which had been pointed out already in 1859 by Kirchhoff. The amount of energy emitted by a body is determined by the high-frequency spectrum of standing electromagnetic waves and that spectrum should be essentially the same as for the high overtones of an organ pipe, say.