= Hermite oscillator quantization
{c}
{title2=$\Lambda=2n+1,\quad n=0,1,2,\ldots$}
A solution of $b''+(\Lambda-\xi^2)b=0$ decaying at both real infinities has $b=e^{-\xi^2/2}H_n(\xi)$ with a <Hermite polynomial>. Substitution gives $H''-2\xi H'+(\Lambda-1)H=0$; its power-series recurrence terminates at degree $n$ when $\Lambda=2n+1$. The square-integrable <Hermite functions> give the complete oscillator spectrum, including the ground state $n=0$. This quantization also determines <global modes of a quadratically confined Ginzburg-Landau equation> after removing drift.
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