In mathematics, especially spectral theory, Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace–Beltrami operator.
case) by Hermann Weyl for eigenvalues for the Laplace–Beltrami operator acting on functions that vanish at the boundary of a bounded domain
, of Dirichlet eigenvalues (counting their multiplicities) less than or equal to
[1] In 1912 he provided a new proof based on variational methods.
[2][3] Weyl's law can be extended to closed Riemannian manifolds, where another proof can be given using the Minakshisundaram–Pleijel zeta function.
The Weyl law has been extended to more general domains and operators.
In the development of spectral asymptotics, the crucial role was played by variational methods and microlocal analysis.
The extended Weyl law fails in certain situations.
In particular, the extended Weyl law "claims" that there is no essential spectrum if and only if the right-hand expression is finite for all
If one considers domains with cusps (i.e. "shrinking exits to infinity") then the (extended) Weyl law claims that there is no essential spectrum if and only if the volume is finite.
On the other hand, for the Neumann Laplacian there is an essential spectrum unless cusps shrinks at infinity faster than the negative exponent (so the finiteness of the volume is not sufficient).
Weyl conjectured that where the remainder term is negative for Dirichlet boundary conditions and positive for Neumann.
In 1952, Boris Levitan proved the tighter bound of
Robert Seeley extended this to include certain Euclidean domains in 1978.
[4] In 1975, Hans Duistermaat and Victor Guillemin proved the bound of
[6] This generalization assumes that the set of periodic trajectories of a billiard in
has measure 0, which Ivrii conjectured is fulfilled for all bounded Euclidean domains with smooth boundaries.
Since then, similar results have been obtained for wider classes of operators.