I am starting Shubin's book on partial differential equations. I am just reconciling some basics. First, we define a linear differential operator like this. $$A = \sum_{|\alpha| \le m}a_\alpha(x)D^\alpha$$ Where $D=(D_1,...,D_n)$ and $D_j=-i\partial_j$. It just says $a_\alpha(x)$ is an infinitely differentiable function. OK, so first off I am trying to understand how the Laplace operator fits into this definition? Remember the Laplace operator is defined like this. $$\Delta = \sum_j \frac{\partial^2}{\partial x_j^2}$$
To me it seems that, in order to fit into the general definition above of linear diff. operator, we need to set $a_\alpha(x)=-1$, because $D^2=-\frac{\partial^2}{\partial x_j^2}$. OK but if we look at the definition of $A$, we need to sum over all values of $\alpha$ less than or equal to $m=2$ in this case. So how to we reconcile that with the Laplace operator? Or do we just say that $a_\alpha(x)=0$ for $\alpha = 1$.
A follow-up question is about understanding the total symbol, which introduces a variable $\xi$. $$a(x,\xi) = \sum_{|\alpha| \le m}a_\alpha(x)\xi^\alpha$$ It says in the text that the total symbol is $-\xi^2$ for the Laplace operator, which goes along with what I wrote above, however I don't understand the purpose of this variable $\xi$ and what it represents. It seems in this case we would just have that $\xi=D$.
Anyways as you can see I am still not grasping some basic concepts here. Any help is greatly appreciated.