Weyl law
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 operators in a variety of mathematical spaces. It is a result ...
, Weyl's law describes the asymptotic behavior of eigenvalues of the
Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named ...
. 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 and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is asso ...
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 eigenvalue In mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape. The problem of whether one can hear the shape of a drum is: given the Dirichlet eigenvalues, what features of the shape of ...
s (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.


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 In mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear partial differential equations. This includes gener ...
.


Counter-examples

The extended Weyl law fails in certain situations. In particular, the extended Weyl law "claims" that there is no
essential spectrum In mathematics, the essential spectrum of a bounded operator (or, more generally, of a densely defined closed linear operator) is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, "fails badly to be ...
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 Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of r ...
proved a bound of O(\lambda^\log \lambda). In 1952,
Boris Levitan Boris Levitan (7 June 1914 – 4 April 2004) was a mathematician known in particular for his work on almost periodic functions, and Sturm–Liouville operators, especially, on inverse scattering. Life Boris Levitan was born in Berdyans ...
proved the tighter bound of O(\lambda^) for compact closed manifolds.
Robert Seeley Robert Seeley, also Seely, Seelye, or Ciely, (1602-1668) was an early Puritan settler in the Massachusetts Bay Colony who helped establish Watertown, Wethersfield, and New Haven. He also served as second-in-command to John Mason in the Pe ...
extended this to include certain Euclidean domains in 1978. In 1975,
Hans Duistermaat Johannes Jisse (Hans) Duistermaat (The Hague, December 20, 1942 – Utrecht, March 19, 2010) was a Dutch mathematician. Biography Duistermaat attended primary school in Jakarta, at the time capital of the Dutch East Indies, where his family mo ...
and
Victor Guillemin Victor William Guillemin (born 1937 in Boston) is an American mathematician working in the field of symplectic geometry, who has also made contributions to the fields of microlocal analysis, spectral theory, and mathematical physics. He is a te ...
proved the bound of o(\lambda ^) when the set of periodic bicharacteristics has measure 0. This was finally generalized by
Victor Ivrii Victor Ivrii ( rus, Виктор Яковлевич Иврий), (born 1 October 1949) is a Russian, Canadian mathematician who specializes in analysis, microlocal analysis, spectral theory and partial differential equations. He is a professor a ...
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