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?

  • $\begingroup$ -1 is of top order in $h$... $\endgroup$ – 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 $$ \sigma_{\rm classical} (\Delta + V) = |\xi^2| = a_0 (x,\xi) = \sigma_{\rm semiclassical} (h^2 \Delta + h^2 V) .$$

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

  • $\begingroup$ Salut Jean-Marc et bienvenu a MO. $\endgroup$ – alvarezpaiva Mar 15 '14 at 22:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.