Search results for " 37J30"

showing 2 items of 2 documents

Integrability of the one dimensional Schrödinger equation

2018

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.

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 37J30
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Bi-homogeneity and integrability of rational potentials

2020

Abstract In this paper we consider natural Hamiltonian systems with two degrees of freedom for which Hamiltonian function has the form H = 1 2 ( p 1 2 + p 2 2 ) + V ( q 1 , q 2 ) and potential V ( q 1 , q 2 ) is a rational function. Necessary conditions for the integrability of such systems are deduced from integrability of dominate term of the potential which usually is appropriately chosen homogeneous term of V. We show that introducing weights compatible with the canonical structure one can find new dominant terms which can give new necessary conditions for integrability. To deduce them we investigate integrability of a family of bi-homogeneous potentials which depend on two integer para…

Hamiltonian mechanicsPure mathematicsPolynomialDegree (graph theory)Integrable system010308 nuclear & particles physicsApplied MathematicsHomogeneous potentialsRational functionDifferential Galois theoryIntegrability01 natural sciencesHamiltonian systemsymbols.namesakeQuadratic equationIntegerSpecial functions0103 physical sciencessymbolsMSC 37J30[MATH]Mathematics [math]010306 general physicsAnalysisMathematicsJournal of Differential Equations
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