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The Estimates L1-L∞ for the Reduced Radial Equation of Schrödinger

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Abstract: Estimates of the type L1-L∞ for the Schrödinger Equation on the Lineand on Half-Line with a regular potential V(x),express the dispersive nature of the Schrödinger Equation and are the essentialelements in the study of the problems of initial values, the asymptotic timesfor large solutions and Scattering Theory for the Schrödinger equation andnon-linear in general; for other equations of Non-linear Evolution. In general,the estimates Lp-Lp express the dispersive nature of thisequation. And its study plays an important role in problems of non-linearinitial values; likewise, in the study of problemsnonlinear initial values; see [1] [2] [3]. On the other hand,following a series of problems proposed by V. Marchenko [4], that we will nameMarchenko’s formulation, and relate it to a generalized version of Theorem 1 given in [1], the main theorem (Theorem 1) of this article provides atransformation operator W that transforms the Reduced RadialSchrödinger 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 term of quadratic order of potential V(x) in theasymptotic development towards zero and adds to the potential V(x) a bounded term and a term exponentiallydecrease fast enough in the asymptotic development towards infinity, whichcontinues 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 thetransformation operator ,as in the case of (RSEHL) where they were established in [3]. Finally, as an open question, the possibility ofextending the L1-L∞ estimates for the case (RSEHL), where added to thepotential V(x) an analytical perturbation ismentioned.

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