Scale isometries and free Schrödinger evolution

Apply the scale contract to the operator in [F1]. Taking $r=2$ gives the onto isometry $R=A^{-1}:H_D^{\sigma }\to H_D^{\sigma +2}$. The same contract gives the unitary extensions of $W(t)$ and their commutation with scale powers, hence $RW(t)=W(t)R$. Unitarity gives the displayed norm equality.

On the common core, $J_{\sigma }W(t)=W(t)J_{\sigma }$ by the spectral-scale commutation in [F2]; both sides extend continuously, so the identity holds on $H_D^{\sigma }$. As $J_{\sigma }$ is an onto isometry and the group is strongly continuous on $L^2$ by [F3], $t\mapsto W(t)v_0$ is continuous in $H_D^{\sigma }$. The group law and inverse also extend from the core.

Put $x=J_{\sigma }{v}_0$. On the core and then by continuity, $J_{\sigma -2}{v}_0=A^{-1}x$. The range of $A^{-1}$ is $D(A)=D(I-A)$, so [F3] applied to $I-A$ differentiates $W(t)J_{\sigma -2}{v}_0$ in $L^2$, with derivative $i(I-A)W(t)J_{\sigma -2}{v}_0$. The scale-power isometries also give a bounded map $A:H_D^{\sigma }\to H_D^{\sigma -2}$; since $A^{-1}$ is bounded on $L^2$, the identity map from $H_D^{\sigma }$ to $H_D^{\sigma -2}$ is bounded. Thus $I-A$ defines the stated bounded extension, and intertwining with $J_{\sigma -2}$ identifies the derivative as $i{\mathrm{\Delta }}_{D}W(t)v_0$ in $H_D^{\sigma -2}$. Its continuity follows from step 2.1 and boundedness of ${\mathrm{\Delta }}_D$.

Multiplying the derivative identity by $i$ yields $i{\mathrm{\partial }}_{t}v+{\mathrm{\Delta }}_{D}v=0$. The group identity gives $v(0)=v_0$. The realization and scale in [F1]–[F2] specify the homogeneous Dirichlet evolution; no additional classical trace is needed for this scale-valued equation. Together with steps 1.1 and 2.1 this proves all assertions.