Weyl's Law
   HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, especially
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operator (mathematics), operators in a variety of mathematical ...
, Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace–Beltrami operator. This description was discovered in 1911 (in the d=2,3 case) by
Hermann Weyl Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
for eigenvalues for the Laplace–Beltrami operator acting on functions that vanish at the boundary of a bounded domain \Omega \subset \mathbb^d. In particular, he proved that the number, N(\lambda), of Dirichlet eigenvalues (counting their multiplicities) less than or equal to \lambda satisfies : \lim_ \frac = (2\pi)^ \omega_d \mathrm(\Omega) where \omega_d is a volume of the unit ball in \mathbb^d. In 1912 he provided a new proof based on variational methods. Weyl's law can be extended to closed
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s, where another proof can be given using the
Minakshisundaram–Pleijel zeta function The Minakshisundaram–Pleijel zeta function is a zeta function encoding the eigenvalues of the Laplacian of a compact Riemannian manifold. It was introduced by . The case of a compact region of the plane was treated earlier by . Definition For ...
.


Generalizations

The Weyl law has been extended to more general domains and operators. For the Schrödinger operator : H=-h^2 \Delta + V(x) it was extended to : N(E,h)\sim (2\pi h)^ \int _ dx d\xi as E tending to +\infty or to a bottom of essential spectrum and/or h\to +0. Here N(E,h) is the number of eigenvalues of H below E unless there is essential spectrum below E in which case N(E,h)=+\infty. In the development of spectral asymptotics, the crucial role was played by variational methods and microlocal analysis.


Counter-examples

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 E. 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. However for the Dirichlet Laplacian there is no essential spectrum even if the volume is infinite as long as cusps shrinks at infinity (so the finiteness of the volume is not necessary). 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 conjecture

Weyl conjectured that : N(\lambda)= (2\pi)^\lambda ^ \omega_d \mathrm (\Omega)\mp \frac (2\pi)^\omega_\lambda ^\mathrm (\partial \Omega) +o (\lambda ^) where the remainder term is negative for Dirichlet boundary conditions and positive for Neumann. The remainder estimate was improved upon by many mathematicians. In 1922, Richard Courant proved a bound of O(\lambda^\log \lambda). In 1952, Boris Levitan proved the tighter bound of O(\lambda^) for compact closed manifolds. Robert Seeley extended this to include certain Euclidean domains in 1978. In 1975, Hans Duistermaat and Victor Guillemin proved the bound of o(\lambda ^) when the set of periodic bicharacteristics has measure 0. This was finally generalized by Victor Ivrii in 1980.Second term of the spectral asymptotic expansion for the Laplace–Beltrami operator on manifold with boundary. Functional Analysis and Its Applications 14(2):98–106 (1980). This generalization assumes that the set of periodic trajectories of a billiard in \Omega 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.


References

{{DEFAULTSORT:Weyl law Partial differential equations Spectral theory