Dichotomy of WKB-solutions of discrete Schrödinger equation (Q882661)

Asymptotic properties of solutions of the three-term recurrence relation \[ u_{n+1}+u_{n-1}-q_nu_n=0,\quad q_n>0, \tag{*} \] are investigated. It is shown that under the condition \(\sum^{\infty}\frac{1}{q_nq_{n+1}}<\infty\) equation (*) possesses a pair of linearly independent solutions given by the asymptotic formulas \(u_n^+\sim \prod_{k=1}^{n-1}q_k\), \(u_n^-\sim \prod_{k=1}^{n-1}q^{-1}_k\) as \(n\to\infty\). Moreover, if \(q_n>2\) and \(\sum^{\infty}(q_{n-1}q_n^2q_{n+1})^{-1}<\infty\), it holds \(u_n^+\sim \prod_{k=1}^{n-1} \left(q_k- \frac{1}{q_{k-1}}\right)\), \(u_n^-\sim \prod_{k=1}^{n-1}\left(q_k-\frac{1}{q_{k+1}}\right)^{-1}\). The relationship of these formulas to the Riccati equation and continuous fractions associated with (*) is also discussed.