I am looking for an argument or reference proving that for $s>0$ $$ \| e^{-i\Delta t}\psi\|_{H^s(\mathbb{R}^d)}=\|\psi\|_{H^s(\mathbb{R}^d)} $$ where $e^{-i\Delta t}$ is the solution map for the time dependent (free) Schrödinger equation, $$ i\partial_t\psi=\Delta\psi, $$ defined via the functional calculus.
This was mentioned in a talk I attended and I am sure it is a standard fact, but I haven't found a suitable reference. Thank you!
Since $|e^{-it|\xi|^2}|=1,$ it follows that
$$\|e^{-i\Delta t}\psi\|_{H^s(\mathbb{R}_x^d)}=\|\langle \xi\rangle^s e^{-it|\xi|^2}\hat{\psi}\|_{L^2(\mathbb{R}_\xi^d)}=\|\langle \xi\rangle^s \hat{\psi}\|_{L^2(\mathbb{R}_\xi^d)}=\|\psi\|_{H^s(\mathbb{R}_x^d)}.$$