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 (1)
| Label | Occurrences |
|---|---|
| Borg–Marchenko theorem canonical | 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
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| hasVersion |
Borg theorem
NERFINISHED
ⓘ
Marchenko theorem NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
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
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Instruction
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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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.