Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4613654 | Journal of Differential Equations | 2006 | 17 Pages |
Abstract
We consider the following question: given AâSL(2,R), which potentials q for the second order Sturm-Liouville problem have A as its Floquet multiplier? More precisely, define the monodromy map μ taking a potential qâL2([0,2Ï]) to μ(q)=ΦË(2Ï), the lift to the universal cover G=SL(2,R)Ë of SL(2,R) of the fundamental matrix map Φ:[0,2Ï]âSL(2,R),Φ(0)=I,Φâ²(t)=(01q(t)0)Φ(t). Let H be the real infinite-dimensional separable Hilbert space: we present an explicit diffeomorphism Ψ:G0ÃHâH0([0,2Ï]) such that the composition μâΨ is the projection on the first coordinate, where G0 is an explicitly given open subset of G diffeomorphic to R3. The key ingredient is the correspondence between potentials q and the image in the plane of the first row of Φ, parametrized by polar coordinates, which we call the Kepler transform. As an application among others, let C1âL2([0,2Ï]) be the set of potentials q for which the equation âuâ³+qu=0 admits a nonzero periodic solution: C1 is diffeomorphic to the disjoint union of a hyperplane and Cartesian products of the usual cone in R3 with H.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Dan Burghelea, Nicolau C. Saldanha, Carlos Tomei,