On the Growth Properties of Solutions for a Generalized Bi-Axially Symmetric Schr\"{o}dinger Equation

Abstract:

      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.