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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.