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

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 $$ \sigma_{\rm classical} (\Delta + V) = |\xi^2| = a_0 (x,\xi) = \sigma_{\rm semiclassical} (h^2 \Delta + h^2 V) .$$

For the semiclassical quantization, three nice references are the books by Dimassi-Sjöstrand, A. Martinez or M. Zworski.