Let the $V_n$ be iid random variables with $P(V_n=\pm 1)=1/2$. Almost surely, the discrete Schrodinger operator $$ (Hu)_n = u_{n+1} + u_{n-1} + V_n u_n $$ on $\ell^2(\mathbb Z)$ has dense pure point spectrum (aka Anderson localization). I don't think (that is, not until someone corrects me) anyone can describe a concrete $\pm 1$ sequence for which this happens.

Let the $V_n$ be iid random variables with $P(V_n=\pm 1)=1/2$. Almost surely, the discrete Schrodinger operator $$ (Hu)_n = u_{n+1} + u_{n-1} + V_n u_n $$ on $\ell^2(\mathbb Z)$ has dense pure point spectrum (aka Anderson localization). I don't think anyone can describe a concrete $\pm 1$ sequence for which this happens.