On the Growth Properties of Solutions for a Generalized Bi-Axially Symmetric Schr\"{o}dinger Equation
Received:October 30, 2015  Revised:June 08, 2016
Key Word: Schro\"{o}dinger equation   scattering potential   Jacobi polynomials   order and type
Fund ProjectL:
 Author Name Affiliation Devendra KUMAR Department of Mathematics, Faculty of Sciences Al-Baha University, P.O.Box-1988, Alaqiq, Al-Baha-65431, Saudi Arabia, K.S.A. Payal BISHNOI Department of Mathematics, M.M.H. College, Ghaziabad (U.P.), India Mohammed HARFAOUI University Hassan II-Casablanca, Laboratory of Mathematics, Cryptography and Mechanics, F.S.T, B.O.Box 146, Mohammedia, Morocco
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In this paper, we have considered the generalized bi-axially symmetric Schr\"{o}dinger equation $$\frac{\partial^2\varphi}{\partial x^2}+\frac{\partial^2\varphi}{\partial y^2} + \frac{2\nu} {x}\frac{\partial \varphi} {\partial x} + \frac{2\mu} {y}\frac{\partial \varphi} {\partial y} + \{K^2-V(r)\} \varphi=0,$$ where $\mu,\nu\ge 0$, and $rV(r)$ is an entire function of $r=+(x^2+y^2)^{1/2}$ corresponding to a scattering potential $V(r)$. Growth parameters of entire function solutions in terms of their expansion coefficients, which are analogous to the formulas for order and type occurring in classical function theory, have been obtained. Our results are applicable for the scattering of particles in quantum mechanics.