Tilted Ornstein-Uhlenbeck oscillator transformation (${\gamma }_m=\kappa n^2{\tau }^2/2-m/\tau$)

The weighted strain density in Ornstein-Uhlenbeck multiplicative amplification evolves under ${\mathcal{L}}_{n}p=(\kappa /2)p^{\prime \prime }+{\tau }^{-1}(\sigma p{)}^{\prime }+n\sigma p$. Setting $p=e^{-{\sigma }^2/(2\kappa \tau )}\psi$ eliminates its first derivative. Shifting and scaling to $x=(\sigma -\kappa n{\tau }^2)/\sqrt{\kappa \tau }$ gives ${\psi }_{xx}+[1+\kappa n^2{\tau }^3-2\gamma \tau -x^2]\psi =0$. The quantum harmonic oscillator levels yield the displayed eigenvalues. Its ground state is the dominant positive mode; higher sign-changing modes may occur in a transient expansion.