= Solution
The issue is the <square-integrable> condition on the whole real line, not just decay at one end. Because the <quantum harmonic oscillator> potential is <even>, the even and odd parts of a square-integrable <wavefunction> are themselves square integrable and satisfy the same equation. It is therefore enough to examine each parity sequence separately.
For a nonterminating sequence, put $b_j=a_{2j+p}$, where $p=0$ or $1$. Its <recurrence relation> gives
$$
\frac{b_{j+1}}{b_j}=\frac{4j+2p+1-\epsilon}{(2j+p+2)(2j+p+1)}\sim\frac1j.
$$
Since $\epsilon$ is real for an <energy eigenvalue>, the ratio is eventually positive. After multiplying by one constant phase, the tail coefficients are all positive. Choose $0<\delta<1/2$. For sufficiently large $j$, the ratio is at least $(1-\delta)/(j+1)$, so for some $K>0$ the tail coefficients obey $b_j\geq K(1-\delta)^j/j!$. Consequently, for $\xi>0$, the magnitude of $f$ eventually exceeds a positive multiple of $\xi^p e^{(1-\delta)\xi^2}$ after subtraction of a fixed <polynomial>. Multiplication by $e^{-\xi^2/2}$ leaves exponential growth like $e^{(1/2-\delta)\xi^2}$, incompatible with the <square-integrable> condition. Thus each nonzero parity series must terminate. The parity argument also excludes cancellation between the two nonterminating sequences at both infinities. This proves <Normalization forces Hermite-series termination>.
If the terminating <polynomial> has degree $n$, its top nonzero coefficient requires $2n+1-\epsilon=0$. Conversely, when $\epsilon=2n+1$, choosing the seed of parity $n$ nonzero makes the <recurrence relation> terminate exactly at degree $n$. The opposite seed must be zero, since that sequence does not terminate. Therefore
$$
\boxed{E_n=\left(n+\frac12\right)\hbar\omega,\qquad n=0,1,2,\ldots.}
$$
The resulting <polynomials> are proportional to the physicists' <Hermite polynomials>, and $\psi_n\propto H_n(\xi)e^{-\xi^2/2}$ is normalizable because any <polynomial> times this <Gaussian function> is square integrable. The two parity seeds describe the full solution space of the second-order <differential equation>; the preceding argument therefore exhausts the normalizable <stationary states>. Each <eigenspace> has dimension one. General time-dependent <quantum states> can of course be <superpositions> of these <energy eigenstates>.
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