The Estimates L_(1)-L_(∞) for the Reduced Radial Equation of Schrodinger

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 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.

Asymptotic Study to the N-Dimensional Radial Schrdinger Equation for the Quark-Antiquark System

Here an asymptotic study to the N-dimensional radial Schrdinger equation for the quark-antiquark interaction potential employing asymptotic iteration method via an ansatz to the wavefunction is carried out. The complete energy spectra of the consigned system is obtained by computing and adding energy eigenvalues for ground state, for large " r" and for small " r". From this analysis, the mass spectra of heavy quarkonia is derived in three dimensions. Our analytical and numerical results are in good correspondence with other experimental and theoretical studies.

Static pressure redistribution mechanism of non-axisymmetric endwall based on radial equilibrium

Non-axisymmetric endwall contouring has been proved to be an effective flow control technique in turbomachinery.Several different flow control mechanisms and qualitative design strategies have been proposed.The endwall contouring mechanism based on the flow governing equations is significant for exploring the quantitative design strategies of the nonaxisymmetric endwall contouring.In this paper,the static pressure redistribution mechanism of endwall contouring was explained based on the radial equilibrium equation.A quantified expression of the static pressure redistribution mechanism was proposed.Compressor cascades were simulated using an experimentally validated numerical method to validate the static pressure redistribution mechanism.A geometric parameter named meridional curvature(Cme)is defined to quantify the concave and convex features of the endwall.Results indicate that the contoured endwall changes the streamline curvature,inducing a centrifugal acceleration.Consequently,the radial pressure gradient is reformed to maintain the radial equilibrium.The convex endwall represented by positive Cme increases the radial pressure gradient,decreasing the endwall static pressure,while the concave endwall represented by negative Cme increases the endwall static pressure.The Cme helps to establish the quantified relation between the change in the endwall radial pressure gradient and the endwall geometry.Besides,there is a great correlation between the distributions of the Cme and the change in the endwall static pressure.It can be concluded that the parameter Cme can be considered as a significant parameter to parameterize the endwall surface and to explore the quantitative design strategies of the nonaxisymmetric endwall contouring.