TITLE:
The Estimates L1-L∞ for the Reduced Radial Equation of Schrödinger
AUTHORS:
Herminio Blancarte
KEYWORDS:
The Schrödinger Equation on the Half-Line, Reduced Radial Equation of Schrödinger, Conditions Sufficient to Establish the Uniqueness of the Potential and Boundary Conditions Are Named the Generalized Theorem 1, The Marchenko’s Formulation, Reduction of Estimates L1-L∞ for the Reduced Radial Equation of Schrôdinger to Equation on Half-Line
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.9 No.5,
May
29,
2019
ABSTRACT: Estimates of the type L1-L∞ for the Schrödinger Equation on the Line
and on Half-Line with a regular potential V(x),
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 [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 Wthat transforms the Reduced Radial
Schrödinger Equation (RRSE) (whose main characteristic is the addition a singular term of quadratic order to a regular potential V(x)) in the Schrödinger Equation on Half-Line (RSEHL) under W.
That is to say; Weliminates 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 L1-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 L1-L∞ estimates for the case (RSEHL), where added to the
potential V(x) an analytical perturbation is
mentioned.