The Estimates <i>L</i>₁-<i>L</i>_&#8734; for the Reduced Radial Equation of Schrödinger

Abstract

Estimates of the type L 1 - L ∞ for the Schr &#246; dinger Equation on the Line and on Half-Line with a regular potential V (x) , express the dispersive nature of the Schr &#246; dinger 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 Schr &#246; dinger equation and non-linear in general; for other equations of Non-linear Evolution. In general, the estimates L p - L p' 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 [1] [2] [3] . On the other hand, following a series of problems proposed by V. Marchenko [4] , that we will name Marchenko’s formulation, and relate it to a generalized version of Theorem 1 given in [1] , the main theorem ( Theorem 1) of this article provides a transformation operator W that transforms the Reduced Radial Schr &#246; dinger Equation ( RRSE ) (whose main characteristic is the addition a singular term of quadratic order to a regular potential V (x) ) in the Schr &#246; dinger Equation on Half-Line ( RSEHL ) under W. That is to say; W eliminates the singular term of quadratic order of potential V (x) in the asymptotic development towards zero and adds to the potential V (x) a bounded term and a term exponentially decrease fast enough in the asymptotic development towards infinity, which continues guaranteeing the uniqueness of the potential V (x) in the condition of the infinity boundary. Then the L 1 - L ∞ estimates for the ( RRSE ) are preserved under the transformation operator , as in the case of ( RSEHL ) where they were established in [3] . Finally, as an open question, the possibility of extending the L 1 - L ∞ estimates for the case ( RSEHL ), where added to the potential V (x) an analytical perturbation is mentioned.