On bounded solutions of a nonlinear Schrödinger equation on the half-line (Q542154)

The following one-dimensional Schrödinger equation \[ \psi''(x) = \left( 1 - \frac{\varphi(x)}{x} \right) \psi(x), \quad 0<x<+\infty \] with the boundary conditions \[ \psi(0)=0, \quad \psi(+\infty) = 0 \] is considered, where \(\varphi(x)\) depends nonlinearly on \(\psi(x)\) via the equation \[ \varphi''(x) = - \frac{\psi^2(x)}{x}\,, \quad 0<x<+\infty \] with the boundary conditions \[ \varphi(0)=0, \quad \varphi'(\infty) = 0. \] In this paper, the properties of nonzero solutions are studied.