Borg–Marchenko theorem
E924204
The Borg–Marchenko theorem is a fundamental result in inverse spectral theory that characterizes when a potential in a one-dimensional Schrödinger operator is uniquely determined by its spectral data.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Borg–Marchenko theorem canonical | 2 |
| Borg’s uniqueness result for Sturm–Liouville problems | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11411746 — 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: Borg–Marchenko theorem Context triple: [Gelfand–Levitan theory, relatedTo, Borg–Marchenko theorem]
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A.
Gelfand–Levitan theory
Gelfand–Levitan theory is a foundational framework in inverse spectral theory that reconstructs differential operators or potentials from their spectral data using integral equations.
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B.
Bohr–Courant theorem
The Bohr–Courant theorem is a classical result in analytic number theory describing the value distribution of Dirichlet series, particularly the Riemann zeta function, and serves as a precursor to modern universality theorems such as Voronin’s.
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C.
Paley–Wiener theorem
The Paley–Wiener theorem is a fundamental result in harmonic analysis that characterizes which functions arise as Fourier transforms of compactly supported functions (or distributions), linking analytic properties of entire functions with support properties in the original domain.
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D.
May–Wigner stability theorem
The May–Wigner stability theorem is a result in theoretical ecology and random matrix theory showing that large, complex systems with many random interactions are generically unstable beyond a critical level of complexity.
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E.
Szegő limit theorem
The Szegő limit theorem is a fundamental result in analysis and operator theory that describes the asymptotic behavior of determinants of large Toeplitz matrices in terms of the symbol’s integral.
- 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: Borg–Marchenko theorem Target entity description: The Borg–Marchenko theorem is a fundamental result in inverse spectral theory that characterizes when a potential in a one-dimensional Schrödinger operator is uniquely determined by its spectral data.
-
A.
Gelfand–Levitan theory
Gelfand–Levitan theory is a foundational framework in inverse spectral theory that reconstructs differential operators or potentials from their spectral data using integral equations.
-
B.
Bohr–Courant theorem
The Bohr–Courant theorem is a classical result in analytic number theory describing the value distribution of Dirichlet series, particularly the Riemann zeta function, and serves as a precursor to modern universality theorems such as Voronin’s.
-
C.
Paley–Wiener theorem
The Paley–Wiener theorem is a fundamental result in harmonic analysis that characterizes which functions arise as Fourier transforms of compactly supported functions (or distributions), linking analytic properties of entire functions with support properties in the original domain.
-
D.
May–Wigner stability theorem
The May–Wigner stability theorem is a result in theoretical ecology and random matrix theory showing that large, complex systems with many random interactions are generically unstable beyond a critical level of complexity.
-
E.
Szegő limit theorem
The Szegő limit theorem is a fundamental result in analysis and operator theory that describes the asymptotic behavior of determinants of large Toeplitz matrices in terms of the symbol’s integral.
- F. None of above. chosen
Statements (36)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in inverse spectral theory ⓘ |
| appliesTo |
Sturm–Liouville operator
ⓘ
linked to:
Sturm–Liouville problem
one-dimensional Schrödinger operator ⓘ |
| assumes |
real-valued potential
ⓘ
sufficient regularity of the potential ⓘ |
| characterizes | uniqueness of potential from spectral data ⓘ |
| concerns |
inverse spectral problem
ⓘ
reconstruction of potential ⓘ spectral data of differential operators ⓘ |
| field |
functional analysis
ⓘ
inverse spectral theory ⓘ mathematical physics ⓘ operator theory ⓘ spectral theory ⓘ |
| generalizes |
Borg’s uniqueness result for Sturm–Liouville problems
ⓘ
linked to:
Borg–Marchenko theorem
|
| hasVersion |
Borg theorem
ⓘ
Marchenko theorem ⓘ
linked to:
Marchenko theory
|
| implies | uniqueness of potential for given spectral measure ⓘ |
| influenced |
development of inverse scattering methods
ⓘ
theory of integrable nonlinear equations ⓘ |
| involves |
Schrödinger operator on an interval
ⓘ
boundary conditions ⓘ eigenvalues ⓘ norming constants ⓘ |
| isPartOf | classical results in inverse spectral theory ⓘ |
| namedAfter |
Gunnar Borg
ⓘ
Vladimir Marchenko NERFINISHED ⓘ |
| relates |
boundary spectral data to potential
ⓘ
spectral measure to potential ⓘ |
| states | that a potential is uniquely determined by appropriate spectral data ⓘ |
| topicIn | monographs on inverse spectral and scattering theory ⓘ |
| usedIn |
integrable systems
ⓘ
mathematical analysis of differential equations ⓘ quantum mechanics ⓘ scattering theory ⓘ |
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: Borg–Marchenko theorem Description of subject: The Borg–Marchenko theorem is a fundamental result in inverse spectral theory that characterizes when a potential in a one-dimensional Schrödinger operator is uniquely determined by its spectral data.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Borg–Marchenko theorem