Reducibility of 1-D Quantum Harmonic Oscillator with New Unbounded Oscillatory Perturbations

Abstract

Enlightened by Lemma 1.7 in [25], we prove a similar lemma which is based upon oscillatory integrals and Langer's turning point theory. From it we show that the Schrödinger equationcan be reduced in H 1 (R) to an autonomous system for most values of the frequency vector ω, where Λ ⊂ R \ {0}, |Λ| < ∞ and x := √ 1 + x 2 . The functions a k (θ) and b k (θ) are analytic on T n σ and µ ≥ 0 will be chosen according to the value of β. Comparing with [25], the novelty is that the phase functions of oscillatory integral are more degenerate when β > 1.