For bounded potentials whose spatial derivative grows at most like |x|^δ with δ<1, the 1D quantum harmonic oscillator is reducible in L^2 for a Cantor set of frequencies of asymptotically full measure.
Bambusi, Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations
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On the reducibility of the 1d quantum harmonic oscillator with a quasi-periodic bounded potential
For bounded potentials whose spatial derivative grows at most like |x|^δ with δ<1, the 1D quantum harmonic oscillator is reducible in L^2 for a Cantor set of frequencies of asymptotically full measure.