The Estimates L1-L∞ for the Reduced Radial Equation of Schrödinger
Autor: Herminio Blancarte Suárez
Estimates of the type 1LL ∞− for the Schrödinger Equation on the Line and on Half - Line with a regular potential ( ) Vx, express the dispersive nature of the Schrödinger Equation and are the essential elements in the study of the problems of initial values, the asymptotic times for large solutions and Scattering Theory for the Schrödinger equation and non-linear in general; for other equations of Non-linear Evolution. In general, the estimates Lp Lp′ − express the dispersive nature of this equation. And its study plays an important role in problems of non-linear initial values; likewise, in the study of problems nonlinear initial values; see   . On the other hand, following a series of problems proposed by V. Marchenko , that we will name Marchenko’s formulation, and relate it to a generalized version of Theorem 1 given in , the main theorem ( Theorem 1) of this article provides a transformation operator that transforms the Reduced Radial Schrödinger Equation ( RRSE )
(whose main characteristic is the addition a singular term of quadratic order to a regular potential ( ) Vx) in the Schrödi nger Equation on Half - Line ( RSEHL ) under . That is to say; eliminates the singular term of quadratic order of potential ( ) Vx in the asymptotic development towards zero and adds to the potential ( ) Vx a bounded term and a term exponentially decrease fast enough in the asymptotic development towards infinity, which continues guaranteeing the uniqueness of the potential ( ) Vx in the condition of the infinity boundary. Then the 1LL ∞− estimates for the ( RRSE ) are preserved under the transformation operator , as in the case of ( RSEHL ) where they were established in .
Finally, as an open question, the possibility of extending the 1LL ∞− estimates for the case ( RSEHL ),
where added to the potential ( ) Vx an analytical perturbation is mentioned.