The limit circle and limit point criteria for second-order linear difference equations (Q1767816)

The following second-order difference equation is considered \[ -c_n u_{n+1}- c_{n-1} u_{n-1}+ b_n u_n= \lambda a_n u_n,\quad n\in [0,+\infty),\tag{1} \] where \([0,\infty)= \{n\}^\infty_{n=0}\{b_n\}^\infty_0\) is a real sequence, and the real sequences \(\{c_n\}^\infty_{-1}\) and \(\{a_n\}^\infty_0\) satisfy \(c_n\neq 0\). Equation (1) is called of limit point type if for some \(\lambda\in \mathbb{C}\) there is a solution \(u\) of (1) which is not in \(l^2_a(0,\infty)\); otherwise equation (1) is called of limit circle type, where \(l^2_a(0,\infty)\) is the Hilbert space \[ l^2_a(0,\infty)= \Biggl\{u: u=\{u_n\}^\infty_{-1}\subset\mathbb{C}\text{ and }\sum^\infty_{n=0} a_n|u_n|^2< \infty\Biggr\} \] with the inner product \(\langle u,v\rangle= \sum^\infty_{n=0} a_n\overline v_n u_n\). Several necessary and sufficient conditions are established for limit point and limit circle type.