semiclassical principal symbol

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?

• -1 is of top order in $h$... – Hans Feb 3 '14 at 18:05

If you consider a "classical" pseudo-differential operator $a (x,D)$ (see e.g. Hörmander's book, vol. 3) with a polyhomogeneous symbol, e.g. a polynomial in $\xi$, $a (x,\xi) = \sum_{|\alpha| \leq m} a_{\alpha}(x) \xi^{\alpha}$ for a differential operator, then the principal symbol is indeed the top order part $\sum_{|\alpha| = m} a_{\alpha}(x) \xi^{\alpha}$ (the term of higher degree in $\xi$ in general). In this case, the terms in the expansion of the symbol (or the operator) are ordered according to their degree in $\xi$.
Semi-classical operators are of the form $a(x,hD)$, or more generally $\sum_k h^k a_k (x,hD)$, for some small parameter $h$. The meaning of $a(x,hD)$ is that one replaces $a (x,\xi)$ by $a (x,h \xi)$ in the "classical" definition. In this case, the principal symbol is the top order term with respect to $h$, that is $a_0 (x,\xi)$.
For instance, if $V = V (x)$ is a function (e.g. V = - 1 in your case), $h^2 \Delta + V = a (x,hD)$ with $a (x,\xi) = |\xi|^2 + V (x)$ (which is both the principal and full symbol of the operator).
Classical and semiclassical principal symbols coincide in some cases. For instance, if you consider $$h^2 \Delta + h^2 V = a_0 (x,hD) + h^2 a_2 (x,hD)$$ with $a_0 (x,\xi) = |\xi|^2$ and $a_2 (x,\xi) = V (x)$ (with a bounded potential $V$ say), then $$\sigma_{\rm classical} (\Delta + V) = |\xi^2| = a_0 (x,\xi) = \sigma_{\rm semiclassical} (h^2 \Delta + h^2 V) .$$