semiclassical principal symbol

Question

What is the semiclassical principal symbol $\sigma_h$ of the operator $h^2\Delta-1$ (here $\Delta=-\sum_j\partial^{2}_{x_j}$)? $h^2\Delta-1$ is a second order semiclassical partial differential operator, so it makes sense to me that $\sigma_h(h^2\Delta-1)=|\xi|^2$. But I've read that $\sigma_h(h^2\Delta-1)=|\xi|^2-1$. Since the principal symbol is supposed to be the top order part of the total symbol (as I understand it), why is the $-1$ included in the principal symbol?

Answer 1

It is because the subprincipal symbol has to not only be of smaller order in $|\xi|$ but also has to be of smaller order in $h$. Hence $-1$ cannot be in the subprincipal symbol since it is not $O(h)$.