Hermite oscillator quantization ($\Lambda =2n+1,n=0,1,2,\dots$)

A solution of $b^{\prime \prime }+(\Lambda -{\xi }^2)b=0$ decaying at both real infinities has $b=e^{-{\xi }^2/2}{H}_n(\xi )$ with a Hermite polynomial. Substitution gives $H^{\prime \prime }-2\xi H^{\prime }+(\Lambda -1)H=0$; 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.