6533b7dcfe1ef96bd1272a6b
RESEARCH PRODUCT
Integrability of the one dimensional Schrödinger equation
Thierry Combotsubject
Class (set theory)Integrable systemFOS: Physical sciencesComplex analysisAlgebras01 natural sciencesSchrödinger equationsymbols.namesake[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]0103 physical sciencesBoundary value problem0101 mathematics010306 general physicsGauge field theoryMathematical PhysicsMathematical physicsMathematicsMSC: 34M46 34M50 37J30Liouville equation010102 general mathematicsSpectrum (functional analysis)Operator theory[ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph]Statistical and Nonlinear PhysicsMathematical Physics (math-ph)Differential operatorHamiltonian mechanicssymbols34M46 34M50 37J30description
We present a definition of integrability for the one dimensional Schroedinger equation, which encompasses all known integrable systems, i.e. systems for which the spectrum can be explicitly computed. For this, we introduce the class of rigid functions, built as Liouvillian functions, but containing all solutions of rigid differential operators in the sense of Katz, and a notion of natural boundary conditions. We then make a complete classification of rational integrable potentials. Many new integrable cases are found, some of them physically interesting.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2018-02-01 |