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 W？that 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; W？eliminates the singular
References
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Blancarte, H. (2015) Determination of a Linear Differential Equation on Half-Line and Its Spectral Distribution Function from the Others Related. Differential Equations and Applications, 7, 469-488. https://doi.org/10.7153/dea-07-27
[2]
Blancarte, H. (2018) Determination Two Regular Sturm-Liouville Operators from Two Spectra. On the Determination of the Linear Equation of the Second Order on the Half-Line and Its Antecedents. Editorial Académica Espanola, International Book Market Service Ltd., Member of Omni Scriptum Publishing Group, Schaltungsdienst Lange O.H.G., Berlin.
[3]
Weder, R. (2003) The Estimates for the Schrödinger Equation on the Half-Line. Journal of Mathematical Analysis and Applications, 281, 233-243. https://doi.org/10.1016/S0022-247X(03)00093-3
[4]
Marchenko, V.A. (1986) Sturm-Liouville Operators and Applications. In: Operator Theory: Advances and Applications, Vol. 22, Birkhauser Verlang Basel, Germany. https://doi.org/10.1007/978-3-0348-5485-6
[5]
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[6]
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[7]
Bourgain, J. (1999) Global Solutions of Non-linear Schrödinger Equations. American Mathematical Society, 46, 27. https://doi.org/10.1090/coll/046
[8]
Journe, J.L. and Soffer, A. (1991) Decay Estimates for SCHRODINGER Operators. Communications on Pure and Applied Mathematics, 44, 573-604. https://doi.org/10.1002/cpa.3160440504
[9]
Kato, T. (1966) Perturbation Theory of Linear Operators. Library of Congress Catalog Card Number 66-15274, Springer Verlag, USA.
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