Marchenko theory
E924203
Marchenko theory is a framework in inverse scattering theory that reconstructs potentials of differential operators from scattering data, closely paralleling the Gelfand–Levitan approach.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Marchenko theory canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11411745 — 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: Marchenko theory Context triple: [Gelfand–Levitan theory, relatedTo, Marchenko theory]
-
A.
Marchenko–Pastur law
The Marchenko–Pastur law is a probability distribution that describes the asymptotic eigenvalue spectrum of large random covariance matrices in random matrix theory.
-
B.
Wigner semicircle law
The Wigner semicircle law is a fundamental result in random matrix theory that describes how the eigenvalues of large random symmetric (or Hermitian) matrices are distributed according to a characteristic semicircular density.
-
C.
random matrix theory
Random matrix theory is a branch of mathematics and mathematical physics that studies the statistical properties of matrices with randomly chosen entries, with deep applications to fields such as number theory, quantum chaos, and statistical mechanics.
-
D.
Wigner matrices
Wigner matrices are large random symmetric (or Hermitian) matrices with independent, identically distributed entries (up to symmetry) that serve as a fundamental model in random matrix theory for studying eigenvalue statistics and universal spectral behavior.
-
E.
Tracy–Widom distribution
The Tracy–Widom distribution is a probability distribution that describes the fluctuations of the largest eigenvalue in many classes of large random matrices and appears widely in random matrix theory and related probabilistic limit theorems.
- 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: Marchenko theory Target entity description: Marchenko theory is a framework in inverse scattering theory that reconstructs potentials of differential operators from scattering data, closely paralleling the Gelfand–Levitan approach.
-
A.
Marchenko–Pastur law
The Marchenko–Pastur law is a probability distribution that describes the asymptotic eigenvalue spectrum of large random covariance matrices in random matrix theory.
-
B.
Wigner semicircle law
The Wigner semicircle law is a fundamental result in random matrix theory that describes how the eigenvalues of large random symmetric (or Hermitian) matrices are distributed according to a characteristic semicircular density.
-
C.
random matrix theory
Random matrix theory is a branch of mathematics and mathematical physics that studies the statistical properties of matrices with randomly chosen entries, with deep applications to fields such as number theory, quantum chaos, and statistical mechanics.
-
D.
Wigner matrices
Wigner matrices are large random symmetric (or Hermitian) matrices with independent, identically distributed entries (up to symmetry) that serve as a fundamental model in random matrix theory for studying eigenvalue statistics and universal spectral behavior.
-
E.
Tracy–Widom distribution
The Tracy–Widom distribution is a probability distribution that describes the fluctuations of the largest eigenvalue in many classes of large random matrices and appears widely in random matrix theory and related probabilistic limit theorems.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
inverse scattering method
ⓘ
mathematical theory ⓘ |
| applicableTo | short-range potentials ⓘ |
| appliesTo |
Schrödinger operators
NERFINISHED
ⓘ
differential operators ⓘ one-dimensional Schrödinger equation ⓘ radial Schrödinger equation ⓘ |
| approachType | integral-equation approach ⓘ |
| assumes | knowledge of scattering matrix ⓘ |
| basedOn | scattering data ⓘ |
| characteristicEquation | Marchenko equation NERFINISHED ⓘ |
| concerns |
recovery of potential from scattering phase shifts
ⓘ
self-adjoint Schrödinger operators ⓘ |
| developedBy | Vladimir A. Marchenko NERFINISHED ⓘ |
| field |
inverse scattering theory
ⓘ
mathematical physics ⓘ partial differential equations ⓘ spectral theory ⓘ |
| frameworkFor |
existence results in inverse scattering
ⓘ
uniqueness results in inverse scattering ⓘ |
| goal | reconstruction of potentials from scattering data ⓘ |
| historicalPeriod | 20th century ⓘ |
| influenced | development of inverse scattering transform ⓘ |
| input |
bound state energies
ⓘ
norming constants ⓘ reflection coefficient ⓘ |
| mathematicalTool |
Fredholm integral equations
NERFINISHED
ⓘ
transformation operators ⓘ |
| output | potential of a differential operator ⓘ |
| relatedConcept |
Jost solutions
NERFINISHED
ⓘ
scattering phase shift ⓘ spectral measure ⓘ |
| relatedTo |
Borg–Marchenko theorem
NERFINISHED
ⓘ
Gelfand–Levitan theory NERFINISHED ⓘ |
| requires | analytic properties of scattering data ⓘ |
| similarTo | Gelfand–Levitan approach NERFINISHED ⓘ |
| studies | inverse scattering problems ⓘ |
| usedIn |
integrable systems
ⓘ
inverse spectral problems ⓘ quantum scattering theory ⓘ soliton theory ⓘ |
| usedTo |
construct reflectionless potentials
ⓘ
derive soliton solutions via inverse scattering ⓘ |
| uses | Marchenko integral equation NERFINISHED ⓘ |
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: Marchenko theory Description of subject: Marchenko theory is a framework in inverse scattering theory that reconstructs potentials of differential operators from scattering data, closely paralleling the Gelfand–Levitan approach.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.