The Schrödinger equation along curves and the quantum harmonic oscillator

We consider the Schrödinger equation associated to the harmonic oscillator, i∂tu=Hu, where H=−Δ+2|x|, with initial data in the Sobolev space Hs(Rd). With d=2 and s>3/8, we prove almost everywhere convergence of the solution to its initial data as time tends to zero, which improves a result of Yajima (1990) [30]. To this end, we consider almost everywhere convergence for the free Schrödinger along curves. As it turns out, these problems are more or less equivalent to that of the free Schödinger equation along vertical lines. Our results are obtained by showing the equivalence of related maximal estimates.