 Solution

A nonzero polynomial solution of degree $n$ has $a_n\ne 0$ but $a_{n+2}=0$. The recurrence therefore forces $2n+1-ϵ=0$. Since $ϵ=2E/(\hslash \omega )$,Conversely, for this value of $ϵ$, choose only the parity sequence containing degree $n$ and set the other initial coefficient to zero. The recurrence terminates at $n$, producing a nonzero polynomial times the decaying Gaussian. Thus every displayed energy occurs under the supplied normalizability criterion.