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• W2810135566 abstract "Univariate Schrodinger equation under physical unit free enti% ties can be given as follows % %-------------------------------(1)-----------------------------------% begin{eqnarray} -frac{1}{2}frac{d^2psi(x,kappa)}{dx^2}+Vleft(kappa x% right)psi(x,kappa)=E(kappa)psi(x,kappa),qquad % xinleft(-infty,inftyright)% label{eq:SchEq} end{eqnarray} %-------------------------------(1)-----------------------------------% where $E$ is the energy parameter of the system while $kappa$ has %  been used to enable us to focus on a one-parameter family of poten%   tial functions. The wave function $psi$ is assumed to be square %    integrable on the real numbers line, even though certain finite or %  semi finite intervals can be taken into consideration in many prac%   tical cases. % Even though (ref{eq:SchEq}) can be solved exactly in very specific % cases, its solution can be numerically or globally approximated thr% ough certain methods in other circumstances. Quite recently, some % of new studies of Group for Science and Methods of Computing (G4S%  &MC), has been launched to develop a somehow new approach we call % lqlq Coordinate Bendingrqrq even though the idea at the back%  ground is not so complicated. This approach is based on the idea %  of changing the independent variable from $x$ to a new coordinate % which is denoted by $u$ through a given relation $u=u(x)$ where %  $u(x)$ is called lqlq Bending Functionrqrq and we are going %  to use two different bending functions: (1) cubic bending function % which is given through % %-------------------------------(2)-----------------------------------% begin{eqnarray} u(x)=x+nu x^3,% label{eq:CubBend} end{eqnarray} %-------------------------------(2)-----------------------------------% (2) inverse cubic bending function given via % %-------------------------------(3)-----------------------------------% begin{eqnarray} x=u(x)+nu u(x)^3% label{eq:InvCubBend} end{eqnarray} %-------------------------------(3)-----------------------------------% where the positive real parameter $nu$ has been used to deal with %  a function family instead of using a single bending possibility. %    $nu$ value can be detemined in accordance with an appropriate de%    mand. Only one of (ref{eq:CubBend}) and (ref{eq:InvCubBend}) shou%  ld be used in a single attempt. The matrix representation of the %    weight function which multiplies the energy parameter can be better % evaluated through truncations in the case of (ref{eq:CubBend}) whi%  le the same object's structure becomes quite simple in the case whe%  re (ref{eq:InvCubBend}) is considered. % Coordinate bending is realized not only through (ref{eq:CubBend}) % or (ref{eq:InvCubBend}). The following unknown mapping should also % be realized at the same time. % %-------------------------------(4)-----------------------------------% begin{eqnarray} psileft(x,kapparight)={u^{prime}(x)}^{frac{1}{2}}% fleft(x,kapparight) label{eq:Psi2f} end{eqnarray} %-------------------------------(4)-----------------------------------% where $nu$ dependence has not been explicitly shown even though % both the bending function and the new unknown function depends % on this parameter. % By using one of these bending functions and (ref{eq:Psi2f}), % (ref{eq:SchEq}) can be cast into another function such that % the potential of the resulting equation changes and a new weight % function appears as a factor to energy constant. It is possible % to seek models where the new potential becomes having a new struc% ture whose corresponding spectrum is well known. Then the eigen% functions of the Schrodinger operator with that potential can % be used as a basis set for getting approximations through matrix % representation truncations. This and some other possibilities %  will be at the focus during the presentation. % Even though there are some other references we find it sufficient % to refer only the following contributed papers of this conference. % begin{thebibliography}{1} %1% bibitem{BKMD} B. Kalay and M. Demiralp, Energy Dependent Coordinate Bending for % Quantum Dynamics of Screened Coulomb Potential Systems, emph{Abs% tracts of ICNPAA 2018, Yerevan, Armenia, 2018}. % %2% bibitem{SBO} S. Bayat Ozdemir, Coordinate Axis Bending in Univariate Schro% dinger Equations, emph{Abstracts of ICNPAA 2018, Yerevan, Armen% ia, 2018}. % end{thebibliography} end{document}" @default.
• W2810135566 created "2018-07-10" @default.
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• W2810135566 date "2018-05-31" @default.
• W2810135566 modified "2023-10-18" @default.
• W2810135566 title "Coordinate bending studies for univariate Schrodinger equation: Cubic and inverse cubic bending functions" @default.
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