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A famous problem of Carleson asks if $f\in H^s(\mathbb{R}^n)$, under what condition of $s$ do we have almost everywhere pointwise convergence of the solution to the Schrodinger's equation

$$iu_t-\Delta u=0, (x,t)\in\mathbb{R}^n\times\mathbb{R}$$

with initial condition $u(x,0)=f(x)$. This has been solved by Du, Guth, Li, Zhang here 1 and here 2. My question is: what about Schrodinger operators with potential term? For example, what can we say about the pointwise convergence of Schrodinger's equation in an EM potential, or harmonic oscillators?

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    $\begingroup$ According to Lucá--Rogers 2017 link.springer.com/article/10.1007/s00220-016-2722-8 (p.342), this problem with the harmonic oscillator potential has been considered by Yajima first. They also state that this quantum harmonic oscillator problem actually turns out to be equivalent to the free one, though. $\endgroup$
    – Giuseppe Negro
    Commented Jun 21, 2023 at 15:36

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