# Proving that a map is differentiable

Let $$H$$ be a self-adjoint operator on $$\mathcal{H}$$, $$\psi\in D(H)$$ and $$\beta\geq 1/2$$. How can I see that $$L \colon \mathbb{R}\to \mathcal{L}(D(\mathcal{N^\beta}),D(\mathcal{N^{\beta-1/2}})), \quad t\mapsto a\left( e^{iHt}\psi\right)$$ is differentiable? Here $$\mathcal{N}$$ denotes the number operator and $$a$$ the annihilation or creation operator. (it should be true for both.)

• Can you add for completeness the linear relations between $a, \mathcal{N}, H$? – DanielC May 31 at 14:09
• I hope i understand you question right: we defined the annihilation and creation operators on the domain of the squareroot of the number operator: $(a^\dagger(\psi),D(\mathcal{N}^{1/2}))$ and $(a(\psi),D(\mathcal{N}^{1/2}))$. – uzizi_1 May 31 at 16:10