= Solution
Start with $u\in H_s$ for some real $s$. The equation becomes
$$
(1+\Delta)u=f+(1-V)u.
$$
Smooth $f$ belongs to every <periodic Sobolev space>, and multiplication by the smooth coefficient preserves $H_s$ by part (c), so the right side lies in $H_s$. The Fourier multiplier $(1+\Delta)^{-1}$ has coefficient $(1+|m|^2)^{-1}$ and maps $H_s$ isometrically to $H_{s+2}$. Thus $u\in H_{s+2}$. Repeat this argument to get $u\in H_{s+2j}$ for every nonnegative integer $j$. For any derivative order $k$, choose $j$ with $s+2j>k+n/2$ and apply part (b). Therefore \b[$\boxed{u\in C^\infty(\mathbb T^n)}$]. This is elliptic bootstrapping directly from the Fourier multiplier, with no assumption that the solution initially has nonnegative Sobolev order.
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