Classical pseudodifferential operators

In order to develop a symbol calculus for Dirac operators and their powers, we shall temporarily restrict our attention to a single chart domain $U \subset M$, over which the cotangent bundle is trivial: $T^*M\bigr\vert _U \isom U \x \bR^n$. If $P$ is an operator on $\Coo(M)$, or more generally, on a space of sections $\Ga(M, E)$ of a vector bundle $E \to M$, and if $\{\phi_1,\dots,\phi_m\}$ is a finite partition of unity in $\Coo(M)$, then $P(f) = \sum_{i,j=1}^m \phi_i P(\phi_j f)$, so we may as well consider operators which are defined on a single chart domain of $M$. At some later stage, we must ensure that the important properties of such operators are globally defined, independently of the choice of local coordinates.

We will work, then, in local coordinates $(x^1,\dots,x^n)$ over a chart domain $U$; the local coordinates of the cotangent bundle $T^*M\bigr\vert _U$ are

$$(x, \xi) = (x^1,\dots,x^n, \xi_1,\dots,\xi_n), \word{where} \xi \in T^*_xM.$$

Let $E \to M$ be a vector bundle of rank $r$. We assume (without loss of generality) that the vector bundle $E$ is also trivial over $U$, so we can identify $\Ga(U,\End E)$ with $U \x M_r(\bC)$.

A differential operator acting on (smooth) local sections $f \in \Ga(U,E)$ is an operator $P$ of the form

$$P = \sum_{\vert\al\vert\leq d} a_\al(x) D^\al, \word{with} a_\al \in \Ga(U,\End E),$$

where we use the notation $D^\al := D_1^{\al_1} \dots D_n^{\al_n}$, and $D_j := -i \del/\del x^j$, the positive integer $d$ is the order of $P$.

The local coordinates allow us to identify $U$ with an open subset of $\bR^n$. The coefficients $a_\al$ are matrix-valued functions $U \to M_r(\bC)$.

By a Fourier transformation, we can write, for $f \in \Coo_c(U,\bR^r)$,
$$\begin{align} Pf(x) &= (2\pi)^{-n} \int_{\bR^n} e^{ix\xi} p(x,\xi) \hat f(\xi... ...n} \iint_{\bR^{2n}} e^{i(x-y)\xi} p(x,\xi) f(y) d^ny d^n\xi, \end{align}$$
where $p(x,\xi)$ is a polynomial of order $d$ in the $\xi$-variable, called the (complete) symbol of $P$. (Clearly, this symbol depends on the choice of local coordinates.) Here

$$K_p(x,y) := (2\pi)^{-n} \int_{\bR^n} e^{i(x-y)\xi} p(x,\xi) f(y) d^n\xi$$ (18)

is the kernel of $P$, as an integral operator: the inverse Fourier transform of $p(x,\xi)$.

For the Dirac operator $\Dslash$, we can use the local expression of the spin connection to write $\Dslash = -i c(dx^j) \nb^S_{\del_j} = -i c(dx^j)(\del_j+\omega_j(x))$, so the corresponding symbol is

$$p(x,\xi) = c(dx^j)(\xi_j - i \omega_j(x)).$$ (19)

This is a first-order polynomial in the $\xi_j$ variables, so that $\Dslash$ is a first order differential operator. The leading term in $p(x,\xi)$ --the part that is homogeneous in $\xi_j$ of degree one-- is $c(dx^j)\xi_j = c(\xi_j dx^j) = c(\xi)$, where $\xi = \xi_j dx^j$ can be regarded as an element of $\sA^1(U)$.

More generally, a pseudodifferential operator $P$ is given locally by an integral of the form (), where the symbol $p(x,\xi)$ need no longer be a polynomial. In that case, we must specify certain classes of symbols for which these integrals make sense.

When $p(x,\xi)$ is a polynomial in $\xi$, of order at most $d$, we can isolate its homogeneous parts:

$$p(x, \xi) = \sum_{j=0}^d p_{d-j}(x, \xi), \words{where} p_{d-j}(x, t\xi) = t^{d-j} p_{d-j}(x, \xi) \word{for} t > 0.$$

We need a formula for the symbol of the composition of two classical pseudodifferential operators (``classical $\Psi$DOs'', for short). It is not clear a priori when and if two such operators are composable: we remit to [Tay], for instance, for the full story on compositions (and adjoints) of classical pseudodifferential operators, and for the justification of the following formula.

If $P$ is a classical $\Psi$DOs of order $d_1$ with symbol $p \in S^{d_1}(U)$, and if $Q$ is a classical $\Psi$DO of orders $d_2$ with symbol $p \in S^{d_2}(U)$, then the symbol $p \circ q$ of the composition $PQ$ lies in $S^{d_1+d_2}(U)$ and its asymptotic development is given by

$$(p \circ q)(x, \xi) \sim \sum_{\al\in\bN^n} \frac{i^{\vert\al\vert}}{\al!} D_\xi^\al p(x,\xi) D_x^\al q(x,\xi).$$ (20)

To find the terms $(p \circ q)_{d_1+d_2-j}(x,\xi)$ of the symbol expansion, one must substitute () for both $p$ and $q$ into the right hand side of () and rearrange a finite number of terms. For the case $j = 0$, one need only use $\al = 0$ --since $D_\xi^\al$ lowers the order by $\vert\al\vert$-- and in particular, the principal symbols compose easily:

$$(p \circ q)_{d_1+d_2}(x,\xi) = p_{d_1}(x,\xi) q_{d_2}(x,\xi).$$

The composition formula is valid for both scalar-valued and matrix-valued symbols, provided the matrix size $r$ is the same for both operators.

Suppose $U$ and $V$ are open subsets of $\bR^n$ and that $\phi \: U \to V$ is a diffeomorphism. If $P$ is a $\Psi$DO over $U$, then $\phi_*P \: f \mapsto P(\phi^* f)\circ \phi^{-1}$ is a $\Psi$DO over $V$, as can be verified by an explicit change-of-variable calculation. If $P$ is classical, then so also is $\phi_*P$. If $p^\phi$ denotes the symbol of $\phi_*P$, we find that the principal symbols are related by

$$p_d(x,\xi) = p^\phi_d(\phi(x), \phi'(x)^{-t}\xi),$$

where $\phi'(x)^{-t}$ is the contragredient matrix (inverse transpose) to $\phi'(x)$.

This is the change-of-variable rule for the cotangent bundle. The conclusion is that, for any scalar $\Psi$DO $P$ that we may be able to define over a compact manifold $M$, the complete symbol $p(x,\xi)$ will depend on the local coordinates for a given chart of $M$, but the leading term $p_d = \sg^P$ will make sense as an element of $\Coo(T^*M)$ --i.e., a function on the total space of the cotangent bundle. (The subleading terms $p_{d-j}(x,\xi)$, for $j \geq 1$, will not be invariant under local coordinate changes.)

When $P$ is defined on sections of a vector bundle $E \to M$ of rank $r$, the principal symbol $\sg^P$ becomes a section of the bundle $\pi^*(\End E) \to T^*M$, i.e., the pullback of $\End E \to M$ via the cotangent projection $\pi\: T^*M \to M$.

For the Dirac operator $\Dslash$, which is a first-order differential operator on $\Ga(M,S)$, we get $\sg^\Dslash \in \Ga(T^*M, \pi^*(\End S))$. From (), we get at once

$$\sg^\Dslash(x,\xi) = c(\xi_j dx^j) = c(\xi).$$

Since taking the principal symbol is a multiplicative procedure, we also obtain

$$\sg^{\Dslash^2}(x,\xi) = (\sg_\Dslash(x,\xi))^2 = c(\xi)^2 = g(\xi,\xi) 1_{2^m}.$$

(Here we use the handy notation $1_r$ for the $r \x r$ identity matrix.) Notice that the principal symbol of $\Dl^S$ is also $g(\xi,\xi) 1_{2^m}$, since $\Dslash^2 - \Dl^S = \quarter s$ is a term of order zero (it is independent of the $\xi_j$ variables), thus $\Dslash^2$ and $\Dl^S$ have the same principal symbol.

Note that $\sg^{\Dslash^2}(x,\xi)$ only vanishes when $\xi = 0$, that is, on the zero section of $T^*M$.

In particular, $\Dslash$, $\Dslash^2$, $\Dl$, $\Dl^S$ are all elliptic differential operators.