Weyl law
E117656
The Weyl law is a fundamental result in spectral theory that describes the asymptotic distribution of eigenvalues of the Laplacian (or similar operators) in terms of the volume of the underlying domain or manifold.
All labels observed (7)
How this entity was disambiguated
This entity first appeared as the object of triple T990133 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Weyl law Context triple: [Hermann Weyl, knownFor, Weyl law]
-
A.
Wigner surmise
The Wigner surmise is an approximate formula in random matrix theory that describes the statistical distribution of spacings between neighboring energy levels in complex quantum systems.
-
B.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
C.
Can one hear the shape of a drum?
"Can one hear the shape of a drum?" is a famous 1966 paper by mathematician Mark Kac that explores whether the geometric shape of a domain can be uniquely determined from the spectrum of its Laplacian, encapsulated in the question of whether one can infer a drum’s shape from the sound it makes.
-
D.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
E.
Wigner distribution function
The Wigner distribution function is a quasi-probability distribution used in quantum mechanics and signal processing to represent quantum states in phase space, often exhibiting non-classical features such as negative values.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Weyl law Target entity description: The Weyl law is a fundamental result in spectral theory that describes the asymptotic distribution of eigenvalues of the Laplacian (or similar operators) in terms of the volume of the underlying domain or manifold.
-
A.
Wigner surmise
The Wigner surmise is an approximate formula in random matrix theory that describes the statistical distribution of spacings between neighboring energy levels in complex quantum systems.
-
B.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
C.
Can one hear the shape of a drum?
"Can one hear the shape of a drum?" is a famous 1966 paper by mathematician Mark Kac that explores whether the geometric shape of a domain can be uniquely determined from the spectrum of its Laplacian, encapsulated in the question of whether one can infer a drum’s shape from the sound it makes.
-
D.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
E.
Wigner distribution function
The Wigner distribution function is a quasi-probability distribution used in quantum mechanics and signal processing to represent quantum states in phase space, often exhibiting non-classical features such as negative values.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
asymptotic formula
ⓘ
mathematical theorem ⓘ result in spectral theory ⓘ |
| appliesTo |
Laplacian on bounded domains in Euclidean space
ⓘ
Laplacian on compact Riemannian manifolds ⓘ self-adjoint elliptic differential operators ⓘ |
| assumes |
compactness of the underlying manifold or boundedness of domain
ⓘ
ellipticity of the operator ⓘ self-adjointness of the operator ⓘ |
| concerns | high-energy asymptotics of the spectrum ⓘ |
| describes |
asymptotic distribution of eigenvalues of elliptic operators
ⓘ
asymptotic distribution of eigenvalues of the Laplacian ⓘ |
| field |
analysis
ⓘ
mathematical physics ⓘ partial differential equations ⓘ spectral theory ⓘ |
| generalizedBy |
Weyl law
self-linksurface differs
ⓘ
surface form:
Weyl law for pseudodifferential operators
Weyl law self-linksurface differs ⓘ
surface form:
Weyl–Hörmander spectral asymptotics
|
| gives | leading term of eigenvalue counting function ⓘ |
| hasApplication |
asymptotics of zeros of zeta functions of differential operators
ⓘ
counting eigenfrequencies of vibrating membranes ⓘ |
| hasConsequence | spectral invariants encode geometric information ⓘ |
| hasForm | N(λ) ~ C·λ^{n/2} as λ → ∞ ⓘ |
| hasLeadingCoefficient | (2π)^{-n} times volume of unit ball in R^n times volume of manifold ⓘ |
| hasRefinement |
Weyl law
self-linksurface differs
ⓘ
surface form:
Weyl law for the spectral function
Weyl law with remainder term ⓘ local Weyl law ⓘ |
| implies | growth rate of eigenvalues of Laplacian ⓘ |
| inspired | later developments in microlocal analysis ⓘ |
| involves |
dimension of the manifold
ⓘ
eigenvalue counting function N(λ) ⓘ principal symbol of the operator ⓘ volume of the manifold ⓘ |
| namedAfter | Hermann Weyl ⓘ |
| originallyFormulatedFor |
Dirichlet problem
ⓘ
surface form:
Dirichlet Laplacian on bounded domains in R^n
|
| relatedTo |
Tauberian theorems
ⓘ
Weyl law self-linksurface differs ⓘ
surface form:
Weyl’s law for the Laplacian on a compact surface
heat trace asymptotics ⓘ spectral geometry ⓘ |
| relates |
eigenvalue counting function to volume of the domain
ⓘ
spectrum of Laplacian to geometry of the manifold ⓘ |
| typeOf | semiclassical asymptotic result ⓘ |
| usedIn |
inverse spectral problems
ⓘ
quantum chaos ⓘ quantum mechanics ⓘ study of heat kernel asymptotics ⓘ |
| yearProved | 1911 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Weyl law Description of subject: The Weyl law is a fundamental result in spectral theory that describes the asymptotic distribution of eigenvalues of the Laplacian (or similar operators) in terms of the volume of the underlying domain or manifold.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Hermann Weyl
this entity surface form:
Weyl’s law
this entity surface form:
Weyl law for the spectral function
this entity surface form:
Weyl–Hörmander spectral asymptotics
this entity surface form:
Weyl law for pseudodifferential operators
this entity surface form:
Weyl’s law for the Laplacian on a compact surface
this entity surface form:
Weyl law for eigenvalues