\documentclass[11pt,twoside]{article} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color} \usepackage[bookmarksnumbered, colorlinks]{hyperref} \usepackage{float} \usepackage{lipsum} \usepackage{afterpage} \usepackage[labelfont=bf]{caption} \usepackage[nottoc,notlof,notlot]{tocbibind} %\renewcommand\bibname{References} \def\bibname{\Large \bf References} \usepackage{lipsum} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \renewcommand{\headrulewidth}{0pt} \fancyhead[LE,RO]{\thepage} \thispagestyle{empty} %\afterpage{\lhead{new value}} \fancyhead[CE]{O. Ozturk} \fancyhead[CO]{A New Perspective via Fractional Calculus} %\topmargin=-1.6cm \textheight 17.5cm% \textwidth 12cm % \topmargin 8mm % \oddsidemargin 20mm % \evensidemargin 20mm % \footskip=24pt % \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} %\theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \renewenvironment{proof}{{\bfseries \noindent Proof.}}{~~~~$\square$} \makeatletter \def\th@newremark{\th@remark\thm@headfont{\bfseries}} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % If you want to insert other packages. Insert them here %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\long\def\symbolfootnote[#1]#2{\begingroup% %\def\thefootnote{\fnsymbol{footnote}}\footnote[#1]{#2}\endgroup} \def \thesection{\arabic{section}} \begin{document} %\baselineskip 9mm %\setcounter{page}{} \thispagestyle{plain} {\noindent Journal of Mathematical Extension \\ Vol. XX, No. XX, (2017), pp-pp (Will be inserted by layout editor)}\\ ISSN: 1735-8299\\ URL: http://www.ijmex.com\\ ‎\vspace*{9mm} ‎ \begin{center} {\Large \bf A New Perspective via Fractional Calculus for the Radial Schr{\"{o}}dinger Equation\\} \let\thefootnote\relax\footnote{\scriptsize Received: April 20, 2017; Accepted: XXXX (Will be inserted by editor)} {\bf Okkes Ozturk$^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\ \vspace{2mm} {\small Bitlis Eren University} \vspace{2mm} \end{center} \vspace{4mm} {\footnotesize \begin{quotation} {\noindent \bf Abstract.} Differintegral theorems are applied to solve some ordinary differential equations and fractional differential equations. By using these theorems, we obtain different results in the fractional differintegral forms. In this paper, we aim to solve the radial Schr{\"{o}}dinger equation under the potential $ V(r)=H/r^{2}-K/r+Lr^{\kappa} $ in $ \kappa=0,-1,-2 $ cases. We also obtain the solutions in the hypergeometric form. \end{quotation} \begin{quotation} \noindent{\bf AMS Subject Classification:} 26A33; 34A08 \noindent{\bf Keywords and Phrases:} Fractional calculus, differintegral theorems, fractional solutions, radial Schr{\"{o}}dinger equation \end{quotation}} \section{Introduction} \label{intro} % It is advised to give each section and subsection a unique label. As is known, the order of derivative consists of an integer in ordinary calculus; differentiation with integer order is always provided for a favorable function. What's interesting is that a function can be held differentiation with any arbitrary order in fractional calculus that is an area of mathematics that grows out of the traditional definitions of the calculus integral and derivative operators. And, same situation is valid for the integration. So, differentiation and integration are generalized such as differintegral in fractional calculus theory. This theory has an important position in areas of science and engineering such as robot technology, PID control systems, Schr{\"{o}}dinger equation, heat transfer, relativity theory, economy, filtration, controller design, mechanics, optics, modelling and so on \cite{Miller, Oldham, Podlubny, Ross, Yilmazer}.\\ Riemann-Liouville differintegral definitions are, respectively, \begin{equation} _aD_{t}^{-\mu}\varphi(t)=[\varphi(t)]_{-\mu}=\frac{1}{\Gamma(\mu)}\int_a^t\frac{\varphi(\omega)}{(t-\omega)^{1-\mu}}d\omega\quad(t>a,\mu>0), \label{1.1} \end{equation} and, \begin{equation} _aD_{t}^{\mu}\varphi(t)=[\varphi(t)]_{\mu}=\frac{1}{\Gamma(k-\mu)}\frac{d^{k}}{dt^{k}}\int_a^t\frac{\varphi(\omega)}{(t-\omega)^{\mu+1-k}}d\omega, \label{1.2} \end{equation} \[ (k-1\leq\mu0) $ is said to be the fractional derivative of $ \varphi(z) $ of order $ \mu $ and, $ \varphi_{\mu}(z)~~(\mu<0) $ is said to be fractional integral of $ \varphi(z) $ of order $ -\mu $, provided that $ \mid \varphi_{\mu}(z) \mid<\infty~~(\mu\in\mathbf{R}) $ \rm\cite{Campos, Nishimoto, Ortigueira}. \end{definition} \begin{lemma}[Linearity] Let $ \varphi(z) $ and $ \psi(z) $ be analytic and single-valued functions. If $ \varphi_{\mu}(z) $ and $ \psi_{\mu}(z) $ exist, then \begin{equation} [\alpha\varphi(z)+\beta\psi(z)]_{\mu}=\alpha\varphi_{\mu}(z)+\beta\psi_{\mu}(z), \end{equation} where $ \alpha $ and $ \beta $ are constants and, $ \mu\in\mathbf{R}, z\in\mathbf{C} $ \rm\cite{Yilmazer}. \end{lemma} \begin{lemma}[Index law] Let $ \varphi(z) $ be an analytic and single-valued function. If $ (\varphi_{\eta})_{\mu}(z) $ and $ (\varphi_{\mu})_{\eta}(z) $ exist, then \begin{equation} (\varphi_{\eta})_{\mu}(z)=(\varphi_{\eta+\mu})(z)=(\varphi_{\mu})_{\eta}(z), \end{equation} where $ \mu,\eta\in\mathbf{R}, z\in\mathbf{C} $ and $ \Big| \frac{\Gamma(\mu+\eta+1)}{\Gamma(\mu+1)\Gamma(\eta+1)} \Big|<\infty $ \rm\cite{Yilmazer}. \end{lemma} \begin{lemma}[Generalized Leibniz rule] Let $ \varphi(z) $ and $ \psi(z) $ be analytic and single-valued functions. If $ \varphi_{\mu}(z) $ and $ \psi_{\mu}(z) $ exist, then \begin{equation} [\varphi(z)\psi(z)]_{\mu}=\sum_{k=0}^{\infty}\frac{\Gamma(\mu+1)}{\Gamma(\mu+1-n)\Gamma(k+1)}\varphi_{\mu-k}(z)\psi_{k}(z), \label{2.3} \end{equation} where $ \mu\in\mathbf{R}, z\in\mathbf{C} $ and $ \Big| \frac{\Gamma(\mu+1)}{\Gamma(\mu+1-n)\Gamma(n+1)} \Big|<\infty $ \rm\cite{Yilmazer}. \end{lemma} \begin{remark} Let $ \vartheta $ be a constant as $ \vartheta\neq0 $. Then \rm\cite{Yilmazer}, \begin{equation} (e^{\vartheta{z}})_{\mu}=\vartheta^{\mu}e^{\vartheta{z}}\quad(\mu\in\mathbf{R}, z\in\mathbf{C}), \label{2.4} \end{equation} \begin{equation} (e^{-\vartheta{z}})_{\mu}=e^{-i\pi{\mu}}\vartheta^{\mu}e^{-\vartheta{z}}\quad(\mu\in\mathbf{R}, z\in\mathbf{C}), \label{2.5} \end{equation} \begin{equation} (z^{\vartheta})_{\mu}=e^{-i\pi{\mu}}\frac{\Gamma(\mu-\vartheta)}{\Gamma(-\vartheta)}z^{\vartheta-\mu}\quad\Big(\mu\in\mathbf{R}, z\in\mathbf{C}, \Big| \frac{\Gamma(\mu-\vartheta)}{\Gamma(-\vartheta)} \Big|<\infty\Big). \label{2.6} \end{equation} \end{remark} \begin{remark} \begin{equation} \Gamma(\mu-k)=(-1)^{k}\frac{\Gamma(\mu)\Gamma(1-\mu)}{\Gamma(k+1-\mu)}\quad(k\in\mathbf{Z}^{+}\cup\{0\}, \mu\in\mathbf{R}). \label{2.7} \end{equation} \end{remark} \begin{theorem}\label{myth1} Let $ \varphi_{-\mu}\neq0 $ where $ \varphi $ is a given function and, $ M(z;m) $ and $ N(z;n) $ be polynomials in $ z $ of degrees $ m $ and $ n $, respectively, given by \begin{equation} M(z;m)=\sum_{k=0}^{m}a_{k}z^{m-k}=a_{0}\prod_{j=1}^{m}(z-z_{j})\quad(a_{0}\neq0, m\in\mathbf{N}), \label{2.8} \end{equation} and, \begin{equation} N(z;n)=\sum_{k=0}^{n}b_{k}z^{n-k}\quad(b_{0}\neq0, n\in\mathbf{N}). \label{2.9} \end{equation} Thus, the nonhomogeneous linear ordinary fractional differintegral equation \[ M(z;m)\chi_{\eta}(z)+\Bigg[\sum_{k=1}^{m}{\mu \choose k}M_{k}(z;m)+\sum_{k=1}^{n}{\mu \choose k-1}N_{k-1}(z;n)\Bigg]\chi_{\eta-k}(z) \] \begin{equation} +{\mu \choose k}n!b_{0}\chi_{\eta-n-1}(z)=\varphi(z)\quad(m,n\in\mathbf{N}, \mu,\eta\in\mathbf{R}), \label{2.10} \end{equation} has a particular solution as follows \begin{equation} \chi(z)=\Bigg[\Bigg(\frac{\varphi_{-\mu}(z)}{M(z;m)}e^{\sigma(z;m,n)}\Bigg)_{-1}e^{-\sigma(z;m,n)}\Bigg]_{\mu-\eta+1}, \label{2.11} \end{equation} \[ (z\in\mathbf{C}\backslash\{z_{1},...,z_{m}\}), \] where for suitable condition, \begin{equation} \sigma(z;m,n)=\int^{z}\frac{N(\zeta;n)}{M(\zeta;m)}d\zeta\quad(z\in\mathbf{C}\backslash\{z_{1},...,z_{m}\}), \label{2.12} \end{equation} confirmed that the second component of Equ. (\ref{2.11}) exists. Moreover, the homogeneous linear ordinary fractional differintegral equation \[ M(z;m)\chi_{\eta}(z)+\Bigg[\sum_{k=1}^{m}{\mu \choose k}M_{k}(z;m)+\sum_{k=1}^{n}{\mu \choose k-1}N_{k-1}(z;n)\Bigg]\chi_{\eta-k}(z) \] \begin{equation} +{\mu \choose k}n!b_{0}\chi_{\eta-n-1}(z)=0\quad(m,n\in\mathbf{N}, \mu,\eta\in\mathbf{R}), \label{2.13} \end{equation} has solutions as follows \begin{equation} \chi(z)=\alpha[e^{-\sigma(z;m,n)}]_{\mu-\eta+1}, \label{2.14} \end{equation} where $ \sigma(z;m,n) $ is given by Equ. (\ref{2.12}) and, $ \alpha $ is an arbitrary constant \rm\cite{Lin}. \end{theorem} \section{Main Results} The radial Schr{\"{o}}dinger equation under the potential $ V(r)=H/r^{2}-K/r+Lr^{\kappa} $ is defined as \cite{Aygun} \begin{equation} R_{2}(r)+\frac{2m}{\hbar^{2}}\Big[E-\frac{H}{r^{2}}+\frac{K}{r}-Lr^{\kappa}-\frac{l(l+1)\hbar^{2}}{2mr^{2}}\Big]R(r)=0, \label{3.1} \end{equation} where $ H,K $ and $ L $ are positive constants. Now, we get \begin{equation} \begin{aligned} & -\lambda^{2}=\frac{2mE}{\hbar^{2}},\quad \textbf{H}=\frac{2mH}{\hbar^{2}},\quad \textbf{K}=\frac{2mK}{\hbar^{2}}, \\ & \textbf{L}=\frac{2mL}{\hbar^{2}},\quad \rho=l(l+1),\quad \tau=\textbf{H}+\rho. \label{3.2} \end{aligned} \end{equation} And, by substituting (\ref{3.2}) into (\ref{3.1}), we have \begin{equation} r^{2}R_{2}(r)-(\lambda^{2}r^{2}-\textbf{K}{r}+\textbf{L}{r^{\kappa+2}}+\tau)R(r)=0. \label{3.3} \end{equation} According to the values of $ \kappa=0,-1,-2 $, Equ. (\ref{3.3}) is given by \begin{equation} r^{2}R_{2}(r)-[(\lambda^{2}+\textbf{L})r^{2}-\textbf{K}{r}+\tau]R(r)=0\quad(\kappa=0), \label{3.4} \end{equation} \begin{equation} r^{2}R_{2}(r)-[\lambda^{2}r^{2}+(\textbf{L}-\textbf{K})r+\tau]R(r)=0\quad(\kappa=-1), \label{3.5} \end{equation} \begin{equation} r^{2}R_{2}(r)-[\lambda^{2}r^{2}-\textbf{K}{r}+(\textbf{L}+\tau)]R(r)=0\quad(\kappa=-2). \label{3.6} \end{equation} In this paper, we use the fractional calculus theorems for (\ref{3.4}), (\ref{3.5}) and (\ref{3.6}) and so, we find particular solutions of the radial Schr{\"{o}dinger equation in the fractional differintegral forms. \begin{theorem}\label{myth2} If $ \mid \varphi_{\mu}(z) \mid\infty~~(\mu\in\mathbf{R}) $ and $ \varphi_{-\mu}\neq0 $, then \begin{equation} Az^{2}\chi_{2}+Bz\chi_{1}+(Dz^{2}+Ez+F)\chi=\varphi, \label{3.7} \end{equation} \[ (A,D\neq0, z\in\mathbf{C}\backslash\{0\}, \chi=\chi(z)), \] has a particular solution such as:\\ $ \chi=z^{\nu}e^{\vartheta z}\Bigg \{\Bigg[A^{-1}z^{-(\mu+1)+\frac{2A\nu+B}{A}}e^{2\vartheta z} $ \begin{equation} .\Big(z^{-(\nu+1)}e^{-\vartheta z}\varphi \Big)_{-\mu}\Bigg]_{-1}z^{\mu-\frac{2A\nu+B}{A}}e^{-2\vartheta z}\Bigg\}_{\mu-1}, \label{3.8} \end{equation} where $ \nu, \vartheta $ and $ \mu $ are in the form: \begin{equation} \nu=\frac{A-B\pm\sqrt{(A-B)^{2}-4AF}}{2A},\quad\vartheta=\pm i\sqrt{\frac{D}{A}}, \label{3.9} \end{equation} and, \begin{equation} \mu=\frac{(2A\nu+B)\vartheta+E}{2A\vartheta}. \label{3.10} \end{equation} Moreover, \begin{equation} Az^{2}\chi_{2}+Bz\chi_{1}+(Dz^{2}+Ez+F)\chi=0, \label{3.11} \end{equation} \[ (A,D\neq0, z\in\mathbf{C}\backslash\{0\}, \chi=\chi(z)), \] has the particular solution in the form: \begin{equation} \chi=\alpha z^{\nu}e^{\vartheta z}\Big(z^{\mu-\frac{2A\nu+B}{A}}e^{-2\vartheta z}\Big)_{\mu-1} \end{equation} where $ \nu $ and $ \vartheta $ are given by Equ. (\ref{3.9}), and $ \mu $ is given by (\ref{3.10}) and, $ \alpha $ is an arbitrary constant \rm\cite{Lin}. \end{theorem} \begin{theorem} According to the expression of Theorem \ref{myth2}, radial Schr{\"{o}}dinger equation in Equ. (\ref{3.4}) has the particular solution in the form: \begin{equation} R(r)=\alpha r^{\nu}e^{\vartheta r}\Big(r^{\mu-2\nu}e^{-2\vartheta r}\Big)_{\mu-1}, \label{3.13} \end{equation} where \[ \nu=\frac{1\pm\sqrt{1+4\tau}}{2},\quad\vartheta=\pm\sqrt{\lambda^{2}+\textbf{L}}, \] and, \[ \mu=\nu+\frac{\textbf{K}}{2\vartheta}. \] \end{theorem} \begin{theorem} According to the expression of Theorem \ref{myth2}, radial Schr{\"{o}}dinger equation in Equ. (\ref{3.5}) has the particular solution in the form: \begin{equation} R(r)=\alpha r^{\nu}e^{\vartheta r}\Big(r^{\mu-2\nu}e^{-2\vartheta r}\Big)_{\mu-1}, \label{3.14} \end{equation} where \[ \nu=\frac{1\pm\sqrt{1+4\tau}}{2},\quad\vartheta=\pm\lambda, \] and, \[ \mu=\nu+\frac{\textbf{K}-\textbf{L}}{2\vartheta}. \] \end{theorem} \begin{theorem} According to the expression of Theorem \ref{myth2}, radial Schr{\"{o}}dinger equation in Equ. (\ref{3.6}) has the particular solution in the form: \begin{equation} R(r)=\alpha r^{\nu}e^{\vartheta r}\Big(r^{\mu-2\nu}e^{-2\vartheta r}\Big)_{\mu-1}, \label{3.15} \end{equation} where \[ \nu=\frac{1\pm\sqrt{1+4(\textbf{L}+\tau)}}{2},\quad\vartheta=\pm\lambda, \] and, \[ \mu=\nu+\frac{\textbf{K}}{2\vartheta}. \] \end{theorem} \begin{theorem} Let $ \mid (r^{\mu-2\nu})_{k} \mid<\infty $ and $ \mid \frac{-1}{2\vartheta r} \mid<1 $~~($ r\neq0, k\in\mathbf{Z}^{+}\cup\{0\} $). The solution in the form: \[ R(r)=\alpha r^{\nu}e^{\vartheta r}\Big(r^{\mu-2\nu}e^{-2\vartheta r}\Big)_{\mu-1}, \] is written as \[ R(r)=r^{\mu-\nu}e^{-\vartheta r}{_{2}F_{0}\Big[1-\mu, 2\nu-\mu; \frac{-1}{2\vartheta r}\Big]} \] where $ _{2}F_{0} $ is the Gauss hypergeometric function. \end{theorem} \begin{proof} By means of Equ. (\ref{2.3}), we have \begin{equation} R(r)=\alpha r^{\nu}e^{\vartheta r}\sum_{k=0}^{\infty}\frac{\Gamma(\mu)}{\Gamma(\mu-k)k!}(r^{\mu-2\nu})_{k}(e^{-2\vartheta r})_{\mu-1-k}. \label{3.16} \end{equation} By using (\ref{2.5}), (\ref{2.6}) and (\ref{2.7}), we rewrite Equ. (\ref{3.16}) as follows: \begin{equation*} \begin{aligned} R(r)=\alpha r^{\mu-\nu}e^{-\vartheta r}(2\vartheta e^{-i\pi})^{\mu-1}\\ & \times\sum_{k=0}^{\infty}\frac{\Gamma(k+1-\mu)}{\Gamma(1-\mu)}\frac{\Gamma(k+2\nu-\mu)}{\Gamma(2\nu-\mu)}\frac{1}{k!}\Big(\frac{-1}{2\vartheta r}\Big)^{k}. \end{aligned} \end{equation*} Then, \[ R(r)=r^{\mu-\nu}e^{-\vartheta r}\sum_{k=0}^{\infty}(1-\mu)_{k}(2\nu-\mu)_{k}\frac{1}{k!}\Big(\frac{-1}{2\vartheta r}\Big)^{k}. \] where $ 1/\alpha=(2\vartheta e^{-i\pi})^{\mu-1} $.\\ Finally, we obtain \[ R(r)=r^{\mu-\nu}e^{-\vartheta r}{_{2}F_{0}\Big[1-\mu, 2\nu-\mu; \frac{-1}{2\vartheta r}\Big]}. \] \end{proof} \section*{Conclusion} In this study, we used fractional calculus theorems for the radial Schr{\"{o}}dinger equation given by the potential $ V(r)=H/r^{2}-K/r+Lr^{\kappa} $. 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