Massive Laplacian isomorphism on Sobolev spaces

For every real $s$,is an isomorphism. Its inverse is the Fourier multiplier $(1+|\xi {|}^2{)}^{-1}$.

For real $m\ne 0$ and constant real $\alpha$, the Fourier symbol of $-\Delta -\alpha \cdot \nabla +m^2$ has modulus at least $|\xi {|}^2+m^2$. Its inverse therefore maps $L^2$ into $H^2$ with norm at most $\max (1,m^{-2})$, uniformly in the drift.