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\begin{document}
\title[Microlocal Techniques]{Microlocal Techniques for Semiclassical
Problems in Geometric Quantization}
% author one information
\author{David Borthwick}
\address{Mathematics Department, University of California, Berkeley CA
94720}
\email{borthwik@math.berkeley.edu}
\thanks{Supported in part by NSF grant DMS9401807 and by an NSF
Postdoctoral Fellowship}
\subjclass{Primary 53C15, 81S10}
\date{February 12, 1997}
\maketitle
%\tableofcontents
\section{Introduction}
Recently the microlocal analysis of Toeplitz operators developed by
Boutet de Monvel, Sj\"ostrand, and Guillemin
\cite{BS}, \cite{BG}, has been used to obtain various semiclassical
results for geometrically quantized compact K\"ahler
manifolds \cite{BMS}, \cite{BPU1}, \cite{BPU2}.
In this paper we will give an expository account of these methods and
results.
Part of our approach will be to work out the analogous theory for
quantization of $\bbC^{n}$.
The theory of \cite{BG} does not apply directly to this case, because of
the noncompactness, but the microlocal structure is the
same.\footnote{This is wellknown, although to my
knowledge the details worked out here do not appear elsewhere
in the literature.}
And the explicitness with which one can write things down in this
case makes the structure of the arguments, and in particular the prominence
of stationary phase, more transparent.
The development of this theory for $\bbC^n$ will also yield an
improvement of a result of Coburn \cite{C},
a theorem connecting the Poisson bracket of two functions
two the semiclassical limit
of the commutator of the associated operators. The class of functions
for which this was proven in \cite{C} constists of trigonometric
polynomials plus
functions with compact support. We are able to enlarge this to the
class of bounded functions with bounded derivatives
(see Section \ref{defest}).
\section{The compact K\"ahler case}\label{kahler}
Let $X$ be a compact K\"ahler manifold, of
real dimension $2n$ ($\bbC P^n$, for example, or
a coadjoint orbit of a compact Lie group), with symplectic form
$\omega$. Let $L\to X$ be a quantizing line bundle, i.e. a
holomorphic hermitian line bundle, the
curvature of whose natural connection is $\omega$.
We form Hilbert spaces by taking holomorphic sections of tensor powers
of $L$,
$$
\hil_k := L^2_{hol}(X, \ltk),
$$
where the inner product is given by integrating the hermitian structure
with respect to the volume form $\omega^n/n!$.
The curvature of the connection induced on $\ltk$ is $k\omega$,
so according to the principles of geometric
quantization, $k = 1/\hbar$. For $k$ sufficiently large, the
dimension of $\hil_k$ is given by the RiemannRoch formula.
For this setup there is a simple way to associate operators to functions,
often referred to as the BerezinToeplitz quantization, after \cite{B1},
\cite{B2}. Let
$$
\Pi_k: L^2(X, \ltk) \to \hil_k
$$
be the orthogonal projection. Given
$f\in C^\infty(X)$ we define $T_k(f) := \Pi_k\circ M(f)$ acting on
$\hil_k$, where $M(f)$ is the operator of multiplication by $f$.
It is not obvious at this point how
microlocal analysis is relevant to the situation, since the $\Pi_k$'s
are projections onto finite dimensional spaces and thus have smooth
kernels. The idea is to roll all the $\hil_k$'s together, using a
standard trick of analysis of operators on vector bundles.
Let $P\subset L^*$ be the unit circle bundle. $P$ is a principal $S^1$
bundle, from which each $\ltk$ may be constructed as an associated
bundle for the appropriate representation of $S^1$. Thus
$C^\infty(X, \ltk)$ may be naturally identified with
$C^\infty(P)_k$, the set of functions on
$P$ satisfying $f(p\cdot e^{i\theta}) = e^{ik\theta} f(p)$.
$P$ inherits a connection form $\alpha$ from
the connection on $L$, and the condition on the curvature of $L$
implies that $d\alpha = \pi^*\omega$.
Thus $\alpha$ is a contact structure, and the corresponding volume
form is $dp = \alpha\wedge (d\alpha)^n/2\pi n!$. The identification
of sections of $\ltk$ with equivariant functions on $P$
extends to an isomorphism
$$
L^2(X, \ltk) \cong L^2(P, dp)_k.
$$
We can now think of $\hil_k \subset L^2(P)$ for all $k$
and so define
$$
\hil = \bigoplus_{k=0}^\infty \hil_k.
$$
This total Hilbert space has a very nice interpretation: $\hil$ is the
Hardy space of $P$, i.e. the $L^2$ space of boundary values of
holomorphic functions on the unit disk bundle of $L^*$.
Another way to phrase this is that
$\hil$ is the $L^2$ completion of the kernel of the boundary
CauchyRiemann operator $\dbarb$ on $P$, which is defined by restriction
of $\dbar$ from the unit disk bundle.
The orthogonal projector $\Pi: L^2(P) \to
\hil$ is called the Szeg\"o projector.
As an operator on $L^2(P)$, the quantization of a function
$f\in \cinf(X)$ is
$$
T_k(f) = \Pi_k M(f) \Pi_k,
$$
where $f$ is pulled back from $X$ to $P$. (This is of course equivalent
to the quantization map given above, but now we regard all operators
as acting on $L^2(P)$.) The full operator $T(f) = \Pi M(f)
\Pi$ is an example of a Toeplitz operator.
A Toeplitz operator of order $m$ is defined to be an operator on $L^2(P)$
which can be written in the form $\Pi Q\Pi$,
where $Q\in \Psi^m(P)$, the space of pseudodifferential operators of
order $m$ on $P$. Denote by
$\calT^m$ the set of Toeplitz operators
of order $m$.
The work of Boutet de Monvel, Sj\"ostrand, and Guillemin, \cite{BS},
\cite{BG}, provides a detailed
picture of the microlocal structure of $\Pi$.
\begin{theorem}\label{wfthm}
\cite{BG} The wave front set of the Schwartz kernel of $\Pi$ is
$$
WF(\Pi) = \{(p,r\alpha_p;\; p,r\alpha_p)\in T^*P\times T^*P;\; p\in P,\; r>0\}
$$
\end{theorem}
\noindent
In fact $\Pi$ is shown to be a Fourier integral operator of Hermite type
and its principal symbol is determined. (Hermite type refers to the fact
that the wave front set is isotropic but not Lagrangian; this makes
the invariant description of the symbol a good deal more complicated.)
For semiclassical purposes, the key point is that the only
singularities of $\Pi$ are large $k$ singularities (since each
$\Pi_k$ was smooth). So analysis of the singularities of $\Pi$
translates into analysis of the semiclassical behavior of $\Pi_k$.
The set of all Toeplitz operators is in fact closed under
composition. This highly nontrivial fact is a consequence of the
following result.
\begin{theorem}\label{copthm}
\cite{BG} Given $A\in \calT^m$,
there exists $Q\in \Psi^m(P)$ satisfying $[\Pi, Q] =
0$ such that $A = \Pi Q\Pi$.
\end{theorem}
Theorem \ref{wfthm} indicates that
$\Pi$ projects wavefront sets into the set
$$
\Sigma = \{(p,r\alpha_p) \in T^*P;\; r>0\}.
$$
Thus as far as the operator $\Pi Q\Pi$ is concerned, the crucial feature
of $Q$ should be the behavior of the total symbol near $\Sigma$.
In fact, the following result shows that the restriction $\sigma(Q)_\Sigma$
gives a meaningful principal symbol for the Toeplitz operator
$\Pi Q\Pi$.
\begin{theorem}\label{symthm} \cite{BG}
If $\Pi Q_1\Pi = \Pi Q_2\Pi$ then $\sigma(Q_1)_\Sigma =
\sigma(Q_2)_\Sigma$. Furthermore, if $Q \in \Psi^m(P)$ and
$\sigma(Q)_\Sigma = 0$, then $\Pi Q\Pi \in \calT^{m1}$.
\end{theorem}
In \cite{G}, Guillemin pointed out that these results lead to a $*$product
on $\cinf(X)$, which we'll now describe.
Let $D_\theta = i\deriv{\theta}$, where $\deriv{\theta}$
is the generator of the $S^1$ action on $P$. Thus $D_\theta \Pi_k = k\Pi_k$.
We can find a parametrix $D_\theta^{1}$ such that $D_\theta^{1}\Pi_k =
k^{1}\Pi_k$ for $k\ge 1$.
\begin{lemma}\label{knorm}
For $Q \in \Psi^m(P)$, there exists a constant $C$ such that for all $k\ge 1$
$$
\norm{\Pi_k Q \Pi_k} \le Ck^m.
$$
\end{lemma}
\begin{proof}
The operator $D_\theta^{m} Q$ is of order zero, and hence bounded
in $L^2(P)$ because $P$ is compact. Let $C = \norm{D_\theta^{m} Q}$.
Since $\Pi_k$ is an orthogonal projection,
$\norm{\Pi_k D_\theta^{m} Q \Pi_k} \le C$. But $\Pi_k D_\theta^{m} Q \Pi_k
= k^{m} \Pi_k Q\Pi_k$ for $k\ge 1$.
\end{proof}
The following theorem (a slight variation on a theorem of \cite{G})
allows us to expand general Toeplitz operators in terms of operators
of the form $T_k(f)$.
\begin{theorem}\label{symexp}
Given $Q \in \Psi^m(P)$ with $[D_\theta, Q]=0$, there exists a series of
functions $f_i \in \cinf(X)$ such that
$$
\Pi_k Q\Pi_k \sim \sum_{j=0}^\infty k^{mj} T_k(f_j),
$$
meaning that if we approximate $\Pi_k Q\Pi_k$ with terms up to $k^{N}$
the norm of the remainder is $O(k^{N1})$.
\end{theorem}
\begin{proof}
The function $\sigma(Q)_\Sigma$
is invariant under the $S^1$ action, of order $m$ in $r$. Hence it
must be of the form $r^m f_m$, where $f_m$ is (the pullback of) a function on
$X$. Now $\Pi Q\Pi  \Pi D_\theta^{m} M(f_m) \Pi \in \calT^{m1}$ and
we simply
iterate the process. Lemma \ref{knorm} gives the bound on the remainder
\end{proof}
The $*$product on $\cinf(X)$ is the formal power series
$$
f * g := \sum_{j=0}^\infty k^{j} B_j(f,g),
$$
where $B_j(f,g)\in\cinf(X)$ is the $j$th function in the expansion of
$T(f) T(g)$ according to Theorem \ref{symexp} (where $T(f) := \Pi M(f) \Pi$).
Associativity of the $*$product follows from the associativity of
Toeplitz operators, and it is clear that $B_0(f,g) = fg$.
Consider $[T(f), T(g)]$. According to Theorem \ref{copthm} there are
operators $Q_f, Q_g$, commuting with $\Pi$, which yield $T(f)$ and $T(g)$.
Then we have
$$
[T(f), T(g)] = \Pi [Q_f, Q_g] \Pi.
$$
Since $\Sigma$ is symplectic, the restriction
$\sigma([Q_f, Q_g]) = i\{\sigma(Q_f), \sigma(Q_g)\}$
to $\Sigma$ is the Poisson bracket
$i\{f,g\}_\Sigma$. The symplectic form on $\Sigma$ is
$dr\wedge d\alpha + r d\alpha$. Recalling that $d\alpha = \pi^*\omega$,
we have
then $\{f,g\}_\Sigma = r^{1} \{f,g\}_\omega$. This means that
$$
B_1(f,g)  B_1(g,f) = i\{f,g\}_\omega,
$$
so indeed we have a $*$product.
Of course, Theorem \ref{symexp} gives more than a purely formal $*$product,
since we have operator norm bounds on remainders.
We can in particular show that the map $f\mapsto T_k(f)$ satisfies two
important properties in Rieffel's definition of deformation quantization
in the the category of $\bbC^*$algebras \cite{R}. This was proven in
\cite{BMS}.
\begin{corollary}\label{defcor}
For $f,g \in\cinf(X)$,
$$
\norm{T_k(f)T_k(g)  T_k(fg)} = O(k^{1}),
$$
and
$$
\norm{k[T_k(f)T_k(g)]  iT_k(\{f,g\})} = O(k^{1}).
$$
\end{corollary}
To close this section, we consider the issue of orderings. Suppose we have a
map $R: \cinf(X) \to \Psi^0(P)$ such that
$\sigma(R_f)_\Sigma = f$. If we then let $T'_k(f) =
\Pi_k R_f \Pi_k$, then all of the properties we have just described remain
the same. This amounts to choosing a different ordering in our quantization
of $f$.
In particular, Tuynman \cite{T} has pointed out that the operator associated
to $f$ by geometric quantization can be written
$\Pi_k (f  \tfrac12 D_\theta^{1} \Delta f) \Pi_k$.
In Section \ref{cn} we will show how to obtain the Weyl quantization by this
technique. In general, semiclassical results will depend on the choice of
ordering, but most formulas can be
translated into a different ordering scheme with simple modifications.
\section{Quantization of $\bbC^n$}\label{cn}
We now turn to the $\bbC^n$ version of the setup just described.
We'll use the symplectic form
$$
\omega = i \sum_{j=1}^n dz_j \wedge d\zbar_j .
$$
This corresponds to the standard symplectic form on $\bbR^{2n}$
under the identification $z_j = (q_j + ip_j)/\sqrt{2}$.
We need to find a holomorphic
hermitian line bundle $L$, whose curvature is given by $\omega$.
Of course, this line bundle will be trivial, so the hermitian structure can
be specified by a positive function $e^{h(z)}$ (the norm of the
constant section 1 at the point $z$).
The curvature for the associated connection will be given by
$2i \del\dbar h$, so we set $h(z) = z^2/2$.
The Hilbert spaces of holomorphic sections of $\ltk$
naturally isomorphic to weighted Bargmann spaces
$$
\hil_k = L^2_{hol}(\bbC^n, e^{kz^2} \>dl_z),
$$
where $dl_z$ is Lebesgue measure $=d^nq\>d^np$.
The principal bundle $P = \bbC^n \times S^1$ inherits the
connection form
$$
\alpha = d\theta + \frac{i}{2} \sum_{j=1}^n (z_j d\zbar_j  \zbar_j dz_j).
$$
Thus the contact volume form is $dp = d\theta\> dl/2\pi$. As in
Section \ref{kahler} we rewrite the Hilbert spaces as subspaces of
$L^2(P)$:
$$
\hil_k = \{F(z,\theta) \in L^2(P): F(z,\theta) = f(z)e^{ik\theta  kz^2/2},
\;f\text{ is holomorphic} \}.
$$
Then we define the total Hilbert space $\hil = \oplus_{k=1}^\infty
\subset L^2(P)$ and orthogonal projector $\Pi: L^2(P) \to \hil$.
Although $\hil$ indeed consists of boundary values of holomorphic
functions on the domain $\{(z,\zeta) \in \bbC^n\times
\bbC: \zeta^2 < e^{z^2}\}$, this $\Pi$ is not a Szeg\"o projector
because the domain is not compact.
The boundary CauchyRiemann operator for this case is
$$
\dbarb f = \sum_j
\left(\frac{\del f}{\del\zbar_j}  \frac{iz_j}{2}
\frac{\del f}{\del\theta} \right) \; d\zbar_j.
$$
Note that $\dbarb f = 0$ has solutions which are not boundary values of
holomorphic functions, e.g. $e^{ik\theta+kz^2/2}$.
But all such solutions grow exponentially so they are not in
$L^2$. We still can write $\hil$ as the closure of
$\ker\dbarb \cap L^2(P)$.
A basis for $\hil_k \subset L^2(P)$ is given by
$$
\phika(z, \theta) = z^\alpha e^{ik\theta  kz^2/2},
$$
where $\alpha \in \bbN^n$ is a multiindex: $z^\alpha =
z_1^{\alpha_1}\dots z_n^{\alpha_n}$.
Using this basis allows us to write $\Pi_k$ explicitly:
\begin{equation*}
\begin{split}
\Pi_k(z,\theta; w, \eta) &= \sum_\alpha
\frac{\phika(z, \theta) \ol{\phika(w,\eta)}}{\norm{\phika}^2}\\
&= \Bigl(\frac{k}{2\pi}\Bigr)^n e^{ik(\theta  \eta)}
e^{kz\cdot \wbar  kz^2/2  kw^2/2}
\end{split}
\end{equation*}
This is just the reproducing kernel on Bargmann space, with a phase
factor.
Using this explicit formula, it is a simple matter to check that
$$
WF(\Pi) = \{(p, r\alpha_p; p, r\alpha_p) \in T^*P\times T^*P; r>0\},
$$
in agreement with Theorem \ref{wfthm}.
As before, let $\Sigma = \{(p, r\alpha_p) \in T^*P; r>0\}$.
Our basic quantization map is $f \mapsto T_k(f)$. Let
$$
\rho_j := T_k(z_j), \qquad \rho_j^* = T_k(\zbar_j).
$$
A simple computation shows that
$$
\rho^*_j \phi(z) e^{ik\theta  kz^2/2} = \frac{1}{k}
\frac{\del \phi}{\del z_j} e^{ik\theta  kz^2/2}.
$$
Thus $\rho$ and $\rho^*$ behave like creation and annihilation operators,
respectively, and
$$
[\rho^*_i, \rho_j] = \frac{1}{k} \delta_{ij}.
$$
The map $T_k$ moves creation operators to the right:
$$
T_k(z^\alpha \zbar^\beta) = (\rho^*)^\beta \rho^\alpha,
$$
an ordering referred to as antiWick, since Wick ordering moves creation
operators to the left. For $f$ polynomial, we can write
$$
T_k(f) = e^{\rho^*\cdot \deriv{\zbar}}\> e^{\rho\cdot\deriv{z}}\> f
\big_{z=\zbar=0}.
$$
On the other hand, the Weyl ordering would symmetric:
$$
W_k(f) := e^{\rho^*\cdot \deriv{\zbar} + \rho\cdot\deriv{z}}\> f
\big_{z=\zbar=0}.
$$
For example, the two quantizations of the harmonic oscillator potential are
$$
T_k(z^2) = \rho^*\cdot \rho, \qquad W_k(z^2) = (\rho^*\cdot\rho +
\rho\cdot\rho^*)/2.
$$
To translate between the two schemes amounts to a simple calculation:
$$
e^{\rho^*\cdot \deriv{\zbar} + \rho\cdot\deriv{z}} =
e^{\rho^*\cdot \deriv{\zbar}} \>e^{\rho\cdot\deriv{z}}\> e^{\frac{1}{2k}
\deriv{z}\cdot \deriv{\zbar}}.
$$
This means that $W_k(f) = \Pi_k (Wf) \Pi_k$ (at least for $f$ polynomial),
where
$$
W = e^{\frac{1}{2} D_\theta^{1} \deriv{z}\cdot \deriv{\zbar}}.
$$
Note that $Wf$ is a pseudodifferential operator on $P$,
with principal symbol given by $f$. One can extend $W$ from polynomials
to a class of smooth functions, but we will not go into the details here.
\section{Deformation estimates}\label{defest}
We want the map $f \mapsto T_k(f)$ to satisfy Rieffel's deformation estimates:
\begin{gather}
\lim_{k\to\infty} \norm{T_k(f)} = \norm{f}_\infty, \label{nlim}\\
\intertext{and}
\lim_{k\to\infty} \norm{k[T_k(f),T_k(g)]  iT_k(\{f,g\})} = 0. \label{clim}
\end{gather}
For the $\bbC^n$ setup we have described, (\ref{nlim}) was proven
in \cite{C} for
all continuous bounded functions, and (\ref{clim}) was proven for a
restricted class of functions as noted in the introduction.
In this section, we will show that various facts from
the theory of \cite{BG}
may be carried over to $\bbC^n$ without difficulty
despite the noncompactness.
We will arrive at a proof of property (\ref{clim}) for functions
in $C^{4n+6}_b(\bbC^n)$, where the subscript $b$ indicates that the function
and its derivatives out to order $4n+6$ are bounded.
Define the symbol class $S^m_l(T^*P)$ to consist of those
$a\in C^{l}(T^*P)$ satisfying
\begin{equation}\label{dadb}
\del^\alpha_u \del^\beta_\eta a(u,\eta) \le C_{\alpha,\beta}
(1 + \eta)^{m\beta} \qquad\text{for } \alpha+\beta\le l,
(u,\eta)\in T^*P.
\end{equation}
Let $\Psi^m_l(P)$ be the class of pseudodifferential
operators whose total symbols are contained in
$S^m_l(T^*P)$. Since $\dim T^*P = 4n+2$, the CalderonVaillancourt
Theorem tells us that any $Q\in \Psi^0_{4n+3}(P)$ is
bounded on $L^2(P)$.
Motivated by \cite{BG} and the method of proof of Corollary \ref{defcor},
our strategy
is to find, for a given $f\in C_b(X)$, the pseudodifferential
operator $Q_f$ for which
$$
[Q_f, \Pi] = 0\qquad\text{and}\qquad \Pi Q_f \Pi = \Pi M(f) \Pi.
$$
All other facts may then be derived from this construction.
First we consider the commutation.
To make $Q_f$ commute with $\Pi$, it is natural to start by solving
the easier problem of $[\dbarb, Q]=0$.
We will adopt $(z,\xi)$ as complex coordinates for
$T^*\bbC^n$. That is, $\xi_j = (\xi^q_j + i\xi^p_j)/\sqrt{2}$.
Note that the standard phase function for a pseudodifferential operator
becomes $2\re z\cdot\xibar$ in this notation.
This means
that $\sigma(\deriv{z_j}) = i\xibar_j$ and $\sigma(\deriv{\zbar_j}) = i\xi_j$.
For future reference, we compute the Fourier transform of a Gaussian
in these coordinates.
\begin{lemma}\label{fgauss}
$$
\int e^{2i\re z\cdot \xibar} z^\alpha e^{\lambdaz^2} dl_z =
\Bigl(\frac{2\pi}{\lambda}\Bigr)^n \>
\Bigl(\frac{i}{\lambda}\xi\Bigr)^\alpha \; e^{\xi^2/\lambda}.
$$
\end{lemma}
Now let $q$ be the total symbol of $Q$.
We'll assume that $[\deriv{\theta}, Q] = 0$, which means that $q$ is
independent of $\theta$. By computing the total symbol of $[\dbarb, Q]$,
we see that
\begin{equation}\label{djqsym}
[\dbarb, Q] = 0 \Longleftrightarrow
\deriv{\zbar_j}q + \frac{i\kappa}{2} \deriv{\xibar_j} q =
0,\quad j =1,\dots,n.
\end{equation}
\begin{lemma}\label{qppqp}
For $Q\in \Psi^0_{4n+3}(P)$, if $[D_\theta, Q] =0$ and $[\dbarb, Q] = 0$
then we have $Q\Pi = \Pi Q\Pi$.
\end{lemma}
\begin{proof}
By CalderonVaillancourt, $Q$ is continuous on
$L^2(P)$. Also, by assumption $\dbarb Qu = Q\dbarb u = 0$ for $u\in\hil$.
Thus $Q$ preserves $\hil$, which is another way of saying
$Q\Pi = \Pi Q\Pi$.
\end{proof}
In our complex notation, $\Sigma$ is the set
$\{(z,\theta;\xi,\kappa)\in T^*P: \xi = i\kappa z/2, \kappa>0\}$.
We therefore expect the condition $\Pi Q\Pi = \Pi M(f)\Pi$ to require that
$q(z,\xi,\kappa) = f(\tfrac{1}{2}z  \tfrac{i}{\kappa}\xi)$ to leading
order in $\kappa$.
It turns out the full expression is given in terms of a heat kernel.
This is very reminiscent to the relation between the Weyl symbol of an
operator on $L^2(\bbR^n)$ and the Wick or antiWick symbol defined
through the Bargmann transform (see \S2.7 of \cite{F}).
Indeed, one should be able to derive $Q_f$ from these known results.
As above, let $C^{l}_b(\bbC^n)$ be the space of bounded functions with
bounded derivatives to order $l$.
\begin{theorem}
For $f\in C^l_b(\bbC^n)$ let $Q_f$ be the pseudodifferential operator on $P$
with total symbol $q_f$, which for $\kappa\ge 1$ is given by
\begin{equation}\label{qfdef}
q_f(z,\xi,\kappa) = (\tfrac{\kappa}{\pi})^n \chi(\kappa) \int
e^{2\kappa z/2  i\xi/\kappa  w^2} f(w) \>dl_w,
\end{equation}
where $\chi\in\cinf(\bbR)$ satisfies $\chi(\kappa) = 0$ for $\kappa<1/2$
and $\chi(\kappa) = 1$ for $\kappa\ge 1$.
Then $Q_f\in \Psi^0_l(P)$ with
$$
[\Pi, Q_f] = 0
\qquad\text{and}\qquad
\Pi Q_f \Pi = \Pi M(f)\Pi.
$$
\end{theorem}
\begin{proof}
The cutoff $\chi$ is needed to make $q_f$ smooth but serves no other
purpose. Indeed, since taking $Q_f \Pi_k$ will set $\kappa = k$, and
$\Pi = \sum_{k\ge 1} \Pi_k$, the choice of $\chi$ has no effect
on the result. Henceforth we will simply drop it from the notation.
By a simple change of variables,
$$
q_f(z,\xi,\kappa) = (\tfrac{\kappa}{\pi})^n \int
e^{2\kappa w^2} f(w + \tfrac{1}{2}z \tfrac{i}{\kappa}\xi) \>dl_w.
$$
This allows us to estimate derivatives of $q_f$ in terms of derivatives
of $f$, so that $f\in C^l_b(\bbC^n)$ implies $q_f\in
S^0_l(T^*P)$. And an application of (\ref{djqsym}) shows that $[D_j,
Q_f] = 0$. Thus $Q_f\Pi = \Pi Q_f \Pi$ from Lemma \ref{qppqp}.
Now we need to show that $\Pi Q_f \Pi = \Pi M(f)\Pi$, which is
equivalent to showing $(\phikb, Q_f\phika) = (\phikb,
f\phika)$, for all $k,\alpha,\beta$. But note that
$$
\Bigl(\deriv{z}\Bigr)^\beta \phika(0) = \beta!\delta_{\alpha,\beta},
$$
which implies $(\phikb, u) = \beta!
(\deriv{z})^\beta u(0)$ for $u\in\hil_k$.
Since $Q_f$ preserves $\hil_k$, it will thus
suffice to compare $(\deriv{z})^\beta Q_f\phika(0)$ to
\begin{equation}\label{dpf}
\Bigl(\deriv{z}\Bigr)^\beta \Pi M(f) \phika(0)
= \Bigl(\frac{k}{2\pi}\Bigr)^n \int (k\zbar)^\beta f(z)
z^\alpha e^{kz^2}\>dl_z
\end{equation}
Using the definition (\ref{qfdef}),
\begin{equation*}
\begin{split}
&\Bigl(\deriv{z}\Bigr)^\beta Q_f\phika(0) = \tfrac{1}{(2\pi)^{2n}}
\left(\deriv{z}\right)^\beta \int e^{2i\re (zw)\cdot \xibar} q_f(z,\xi,k)
w^\alpha e^{kw^2/2} \>dl_\xi\>dl_w \bigg_{z=0} \\
&\quad = (\tfrac{k}{4\pi^3})^n
\int e^{2i\re w\cdot \xibar} (i\xibar + \tderiv{z})^\beta
e^{2kz/2  i\xi  y^2}
f(y) w^\alpha e^{kw^2/2} \>dl_\xi\>dl_w\>dl_y \bigg_{z=0} \\
&\quad = (\tfrac{k}{2\pi^3})^n
\int e^{2i\re w\cdot \xibar} (k\ybar)^\beta e^{2\xi  iky^2/k}
f(y) w^\alpha e^{kw^2/2} \>dl_\xi\>dl_w\>dl_y
\end{split}
\end{equation*}
Performing the $w$ integration using Lemma \ref{fgauss} gives
\begin{equation*}
\begin{split}
\Bigl(\deriv{z}\Bigr)^\beta Q_f\phika(0) &= \tfrac{1}{\pi^{2n}}
\int (k\ybar)^\beta f(y) (\tfrac{2i}{k} \xi)^\alpha
e^{2kz/2  i\xi  y^2} e^{2\xi^2/k} \>dl_\xi\>dl_y. \\
&= \tfrac{1}{\pi^{2n}} \int (k\ybar)^\beta
f(y) (\tfrac{2i}{k} \xi)^\alpha e^{4\xiiky/2^2/k} e^{ky^2}
\>dl_\xi\>dl_y.
\end{split}
\end{equation*}
Now change variablies $\xi\to \xiiky/2$,
$$
\Bigl(\deriv{z}\Bigr)^\beta Q_f\phika(0)
= \tfrac{1}{\pi^{2n}} \int (k\ybar)^\beta
f(y) (y\tfrac{2i}{k} \xi)^\alpha e^{4\xi^2/k} e^{ky^2}
\>dl_\xi\>dl_y.
$$
If we expand $(y\tfrac{2i}{\kappa} \xi)^\alpha$ and integrate over $\xi$
only the $y^\alpha$ term survives (the others are zero by symmetry). So we
finally obtain
$$
\Bigl(\deriv{z}\Bigr)^\beta Q_f\phika(0)
= (\tfrac{k}{2\pi})^{n} \int (k\ybar)^\beta
f(y) y^\alpha e^{ky^2} \>dl_\xi\>dl_y.
$$
Comparing this to (\ref{dpf}), we have shown that
$(\phikb, Q\phika) = (\phikb, f\phika)$ and thus $\Pi Q_f\Pi = \Pi M(f)\Pi$.
To conclude the proof, a simple calculation shows that $(Q_f)^* = Q_{\bar f}$.
This implies
$$
\Pi Q_f = (Q_{\bar f}\Pi)^* = (\Pi M(\bar f) \Pi)^* = \Pi M(f)\Pi = Q_f\Pi.
$$
\end{proof}
\begin{lemma}\label{psym}
The principal symbol of $Q_f$ is
$$
\sigma(Q_f)(z,\xi,\kappa) = f(\tfrac{1}{2}z \tfrac{i}{2\kappa}\xi) \hskip1in
$$
More precisely, for $f\in C^{l}_b(\bbC^n)$,
$$
q_f(z,\xi,\kappa)  f(\tfrac{1}{2}z \tfrac{i}{2\kappa}\xi) \in
S^{1}_{l2}(T^*P)
$$
\end{lemma}
\begin{proof}
The expression,
$$
q_f(z,\xi,\kappa) = (\tfrac{\kappa}{\pi})^n \int
e^{2\kappa w^2} f(w + \tfrac{1}{2}z \tfrac{i}{\kappa}\xi) \>dl_w,
$$
can be estimated for large $\kappa$ by a variation of the stationary
phase argument. To apply this, one needs to cut off the integral for
large $w$, which gives an error term that decays exponentially in
$\kappa$. Neglecting this correction, we have
$$
q_f(z,\xi,\kappa)  f(\tfrac{1}{2}z  \tfrac{i}{\kappa}\xi)
\le C\kappa^{1} \sum_{\alpha = 2} \sup (\del^\alpha f)
$$
For the derivatives, first differentiate inside the integral
and then make a similar estimate. In each case 2 extra derivatives
are required for the estimate.
\end{proof}
\begin{theorem}
For $f,g\in C^{4n+5}_b(\bbC^n)$,
\begin{equation}
\norm{T_k(f)T_k(g)  T_k(fg)} = O(k^{1}). \label{nfg}
\end{equation}
And for $f,g \in C^{4n+6}_b(\bbC^n)$,
\begin{equation}
\norm{k[T_k(f),T_k(g)]  T_k({f,g})} = O(k^{1}). \label{ncfg}
\end{equation}
\end{theorem}
\begin{proof}
To prove (\ref{nfg}) note that
$$
T_k(f)T_k(g)  T_k(fg) = (Q_f Q_g  Q_{fg}) \Pi.
$$
By Lemma \ref{psym}, $Q_fQ_g$ and $Q_{fg}$ have the same principal
symbol, so $Q_f Q_g  Q_{fg} \in \Psi^{1}_{4n+3}(P)$.
This implies $D_\theta(Q_f Q_g  Q_{fg}) \in \Psi^0_{4n+3}(P)$,
where $D_\theta = i\deriv{\theta}$.
By CalderonVaillancourt $D_\theta(Q_f Q_g  Q_{fg})$ is bounded. But
$$
D_\theta(Q_f Q_g  Q_{fg}) \Pi_k = k(T_k(f)T_k(g)  T_k(fg)).
$$
Since $\norm{\Pi_k} = 1$ we see that
$$
k\norm{T_k(f)T_k(g)  T_k(fg)} \le C,
$$
where $C$ is independent of $k$, so (\ref{nfg}) is proven.
The principle is the same for (\ref{ncfg}). With our complex
notation for $T^*P$, we have
\begin{equation*}
\begin{split}
&\sigma([Q_f,Q_g]) = i\bigl\{f(\tfrac{1}{2}z \tfrac{i}{\kappa}\xi),
g(\tfrac{1}{2}z \tfrac{i}{\kappa}\xi)\bigr\}_{\scriptscriptstyle T^*P} \\
&\quad = 2i\sum_j \left( \tderiv{z_j}\tderiv{\bar\eta_j} + \tderiv{\zbar_j}
\tderiv{\eta_j}
 \tderiv{\xi_j}\tderiv{\wbar_j} 
\tderiv{\xibar_j}\tderiv{w_j} \right)
f(\tfrac{1}{2}z \tfrac{i}{\kappa}\xi) g(\tfrac{1}{2}w
\tfrac{i}{2\kappa}\eta)\bigg_{\substack{w=z\\ \eta=\xi}} \\
&\quad = \frac{1}{\kappa} \sum_{j=1}^n \left( \tfrac{\del f}{\del z_j}
\tfrac{\del g}{\del \zbar_j} 
\tfrac{\del f}{\del \zbar_j} \tfrac{\del g}{\del z_j} \right)
\Big_{z/2  i\xi/2\kappa} \\
&\quad = \frac{i}\kappa \{f,g\}\Big_{z/2  i\xi/2\kappa}.
\end{split}
\end{equation*}
So that $\sigma([Q_f,Q_g]) = \kappa^{1} \sigma(Q_{\{f,g\}})$.
Thus $D_\theta [Q_f,Q_g]  Q_{\{f,g\}} \in \Psi^0_{4n+3}(P)$
and the result follows as above.
\end{proof}
It is natural to try to extend these results to more general complex
domains. Indeed, in \cite{BLU} the deformation estimates were proven for
all noncompact Cartan domains. The proof relied heavily on the
symmetry of the domains and required functions to have
compact support. The methods presented above make no use of symmetry, so
one could imagine extending the results to a much broader class of
domain. The key difficulty is to prove estimates for $q_f$ which show
that $Q_f$ and associated operators are bounded. One cannot localize
the problem in the usual manner, because the smooth error terms may be
unbounded. Work to overcome this difficulty is now in progress.
\section{Semiclassical spectral theory}\label{spthy}
If we fix a Hamiltonian function $H\in \cinf(X)$, then we have a
corresponding family of operators
$$
\hk := \Pi_k M(\pi^*H) \Pi_k \text{ on } L^2(P)
$$
Define a set eigenvalues and normalized eigenfunctions:
$$
\hk \psikj = \lkj \psikj, \qquad j\in \bbN.
$$
One of the major goals of semiclassical analysis is to show how the
classical theory (i.e. the Hamiltonian flow) is manifested in the
large $k$ behavior of $\psikj$ and $\lkj$. Let $\phi_\tau$ denote the
Hamiltonian flow of $H$ on $X$.
A standard way to study the semiclassical behavior of the spectrum of $\hk$
is to take a trace weighted near a particular energy.
Let $\varphi$ be a test function.
Fix an energy $E$ and consider the sums
\begin{equation}\label{trace}
\sum_{j} \varphi(k(\lkj  E))
\end{equation}
and
\begin{equation}\label{husimi}
\sum_{j} \varphi(k(\lkj  E)) \> \psikj(p_1)\> \ol{\psikj(p_2)},
\qquad p_1, p_2 \in P.
\end{equation}
The effect of the test function is to restrict to a range of energies near
$E$, whose size is $O(k^{1})$. The goal is to produce large $k$ asymptotic
expansions for these expressions. Expansions of traces of the form
(\ref{trace}) have now been proven in various settings. The more unusual
localized form (\ref{husimi}) is a K\"ahler version of expressions
whose asymptotic expansions were studied in \cite{PU2}.
For $X$ compact K\"ahler, large $k$ asymptotic expansions were
given for these sums in \cite{BPU2}
(with restrictions on the regularity of $\phi_\tau$ and with the
Fourier transform of $\varphi$ compactly supported).
The trace (\ref{trace}) has an expansion whose terms correspond to fixed
point sets in the energy surface $H^{1}(E)$ of the map $\phi_\tau$ for
some value $\tau$. For instance, the $\tau = 0$ fixed point set is the whole
energy surface, and the leading contribution is $Ck^{n1}$ times
the volume of $H^{1}(E)$. (When $E$ is a regular value of $H$ the energy
surface inherits a natural volume form from the Liouville form.)
And the localized trace (\ref{husimi}) decays rapidly unless both points are
on the energy surface and connected by a finite time classical path.
Each classical path connecting the points gives rise to a term in the
expansion.
To illustrate this behavior, we turn the classical example of a harmonic
oscillator. Take $X =\bbC^n$ with the quantization setup as in Section
\ref{cn}, and let $H(z) = z^2$. The
basis $\{\phika\}$ conveniently provides the set of eigenstates:
$$
\hk \phika = \tfrac{1}{k} (\alpha+n) \phika.
$$
(See the discussion of ordering in Section \ref{cn}.)
For simplicity, take $\varphi$ supported in $(n1/2,n+1/2)$, with
$\varphi(n) = 1$. And assume that $E$ is an integer.
Then (\ref{trace}) becomes
$$
\sum_{\alpha} \varphi(k^{1}(\lambda_{k,\alpha}  E)) =
\#\{\alpha\in \bbN^n: \alpha = kE \} = \binom{n+kE1}{n1}
$$
Applying Stirling's formula gives the asymptotic formula
$$
\sum_{\alpha} \varphi(k^{1}(\lambda_{k,\alpha} E))
= \frac{(kE)^{n1}}{(n1)!} + O(k^{n2})
$$
This calculation agrees with the result quoted above, the
canonical volume of $H^{1}(E)$ being $\pi^n E^{n1}/(n1)!$.
To compute the localized trace (\ref{husimi}), we must normalize $\phika$
by a factor of $\sqrt{k^{n+\alpha}/ (2\pi)^n \alpha!}$.
Then we have
\begin{equation*}
\begin{split}
&\sum_{\alpha} \varphi(k^{1}(\lambda_{k,\alpha}  E)) (\tfrac{k}{2\pi})^n
\tfrac{k^{\alpha}}{\alpha!} \phika(z,0) \ol{\phika(w,0)}\\
&\qquad = \sum_{\alpha = kE} (\tfrac{k}{2\pi})^n
\tfrac{k^{\alpha}}{\alpha!}
z^\alpha \wbar^\alpha e^{kz^2/2kw^2/2}\\
&\qquad = (\tfrac{k}{2\pi})^n k^{kE} \tfrac{1}{(kE)!} (z\cdot \wbar)^{kE}
e^{kz^2/2kw^2/2}.
\end{split}
\end{equation*}
By Stirling's formula this expression is
$$
\sim \frac{k^{n1/2}}{\pi^n \sqrt{2\pi E}}
\left(\frac{z\cdot\wbar}{E}\right)^{kE} e^{kz^2/2kw^2/2 + kE}
$$
for large $k$.
Two conditions have to be met for this expression not to decay exponentially
for large $k$. Namely $z^2 = w^2 = E$ (the points must lie on the
energy surface) and $z = e^{i\theta}w$ for some $\theta$ (the points must
be connected by a classical trajectory). When these conditions are satisfied
the resulting order is $k^{n1/2}$.
Note that for both cases the asymptotics resulted from an application of
Stirling's formula. When one considers that Stirling's formula
can be obtained through stationary phase approximation, the method
used here is really quite close to that of \cite{BPU2}.
The proofs in \cite{BPU2} are based on the following strategy, which
originated in \cite{GU}. Suppose
you want to know the large $k$ behavior of quantities $A_k$. Define
$$
\Upsilon(\theta) = \sum_k A_k e^{ik\theta}.
$$
If $\Upsilon$ turns out to be smooth,
then the $A_k$'s must decay rapidly. And if $\Upsilon$ has singularities,
one can translate precise knowledge of these singularities
into knowledge of the large $k$ behavior of the $A_k$.
The trick then is to write $\Upsilon$ as a composition of distributions,
in such a way that one can trace the singularities.
To analyze (\ref{trace}) or (\ref{husimi}), the appropriate
$\Upsilon$ is written as a composition of kernels
of Hermite Fourier integral operators. The necessary composition theorems
were worked out in \cite{BG}.
The kernels of the operators are written locally
as oscillatory integrals of a particular form, and the determination
of singularities in the composition basically reduces to a stationary
phase approximation.
For the case of the localized trace formula (\ref{husimi}), $\Upsilon$ can
be constructed from the following operators.
Let
$$
B = e^{t\Pi D_\theta M\Pi},
$$
as an operator from $\cinf(P\times P)\to \cinf(\bbR)$.
Let $\calB$ be the Schwartz kernel of this operator.
Having chosen $\varphi,E,p_1,p_2$, we define
$F: \cinf_0(\bbR\times P\times P)\to \cinf(S^1)$ by
$$
F(f)(\beta) = \int e^{i\kappa(\beta+tE\theta)} \hat\varphi(t)
f(t,p_1\cdot e^{i\theta}, p_2) \>d\kappa\>d\theta\>dt.
$$
To see that the desired $\Upsilon$ is $F(\calB)$, write $\calB$ as a sum
over the basis $\psikj$. (Of course, one needs to check first that $F$ can
be extended to distributions such as $\calB$.)
To see where singularities may occur in $\Upsilon$ is a simple matter of
composing wavefront sets (most of the work comes in computing the
leading behavior).
$F$ is an FIO of standard type, with wave front set
$$
\Gamma \subset T^*S^1 \times T^*\bbR\times T^*P \times T^*P
$$
given by
\begin{multline}\label{gdef}
\Gamma = \{(\theta  tE, J(p_1,\eta_1);\; t, EJ(p_1,\eta_1);\;
p_1\cdot e^{i\theta}, \eta_1; p_2, \eta_2): \\
\theta\in \bbR/2\pi\bbZ, t\in \bbR, \eta_i \in T^*_{p_i}P\},
\end{multline}
where $J = \sigma(D_\theta)$. Because
$\alpha$ is a connection form, $J(p,r\alpha_p) = r$.
The wave front set of $\calB$ is the moment Lagrangian in $T^*\bbR
\times \Sigma\times \Sigma$ (which is isotropic as a subset of
$T^*\bbR \times T^*P \times T^*P$):
\begin{equation}\label{tsdef}
\tilde\Sigma = \{(t, rH(\pi(p));\; \phi^h_t(p)\cdot e^{itH(p)},
r\alpha;\; p, r\alpha): t\in\bbR, p\in P, r>0\}
\end{equation}
where $\phi^h$ is the horizontal lift of $\phi$ from $X$ to $P$.
The wave front set of $\Upsilon$ is contained in the composition
$\Gamma\circ\tilde\Sigma \subset T^*S^1$, where
$\Gamma\circ\tilde\Sigma$ is the set of points $a\in T^*S^1$ for which
there exists $b\in \tilde\Sigma$ such that $(a,b)\in \Gamma$.
>From (\ref{gdef}) and (\ref{tsdef}) we see that
$\Gamma\circ\tilde\Sigma$ consists of points $(\theta,r)\in T^*S^1,
r>0$, such that $H(p_1) = E$ and $p_1 = \phi_\tau(p_2)\cdot e^{i\theta}$
for some $\tau$. So we see that the terms decay rapidly unless these
classical conditions are met.
\section{Semiclassical states}
The WKB method in quantum mechanics gives way of constructing approximate
semiclassical
solutions of Schr\"odinger's equation for some Hamiltonian to a Lagrangian
submanifold $\Lambda$ in phase space (see \cite{BW} for a nice exposition).
The construction requires that $\Lambda$ satisfy a BohrSommerfeld
type condition, reflecting the fact that only certain energies occur quantum
mechanically.
For compact K\"ahler $X$, quantized as in Section \ref{kahler}, the
BohrSommerfeld condition is that $\Lambda$ should have a horizontal lifting
up to $P$. Or equivalently, $\Lambda$ should be given as
$\pi(\tilde\Lambda)$ where
$\tilde\Lambda$ is a Legendrian submanifold of $P$. (Legendrian means simply
$\alpha_{T\tilde\Lambda} = 0$). A simple way to associate states to
$\tilde\Lambda$ (not using WKB)
is to take $\Pi(\delta_\nu)$, where $\delta_\nu$ is the deltafunction
distribution associated to some density $\nu$ on $\tilde\Lambda$.
Since $\Pi$ applied to the deltafunction of a point gives a coherent state,
this essentially amounts to integrating coherent states over a
submanifold of $X$. The lifting to $P$ amounts to a choice of
phase for these states.
The problem addressed in \cite{BPU1} is to relate
the inner products of states constructed in this way to the underlying
Lagrangian submanifolds, in the semiclassical limit.
The results are that the norm of
such a state has an asymptotic expansion with leading term given by the volume
of $\Lambda$ with respect to the chosen density. And the inner product of two
different states has an expansion with terms coming from intersections of
the submanifolds.
We will demonstrate how this comes about in the case of $X = \bbC$.
Our calculations are essentially a repeat of calculations done in
\cite{PU1}. Let $\gamma(t)$ be a smooth regular
closed curve in $\bbC$ with period $2\pi$.
Because we are in dimension two, any such $\gamma$ is a Lagrangian submanifold.
To lift to $P$, we add an angle component $\vartheta(t)$.
The pair $(\gamma, \vartheta)$ will be horizontal if
$\alpha(\gamma', \vartheta') = 0$, or in other words, if
\begin{equation}\label{vart}
\vartheta' = \tfrac{i}{2} (\gamma\gbar'  \gbar \gamma').
\end{equation}
The BohrSommerfeld condition for such a curve is simply $\vartheta(2\pi) =
\vartheta(0)$. From the formula for $\vartheta'$ we see that $\vartheta(2\pi)$
is 2 times the area enclosed by $\gamma$, counted with signs and
multiplicities. Hence this area must be $\pi$ times an integer.
The coherent state is $\Pi(\delta_{w,\eta})(z,\theta) =
\tfrac{k}{2\pi} e^{kz\wbar  kz^2/2kw^2/2}$.
For simplicity, we'll take the density on the curve to be the unit
of arclength. Therefore to the curve $\gamma$, with $\vartheta$
determined by (\ref{vart}), we associate the state
$$
\Phi_{k,\gamma}(z,\theta) = \tfrac{k}{2\pi} e^{kz^2/2}
\oint e^{ik\vartheta(t)} e^{kz\gbar(t)  k\gamma(t)^2/2} \gamma'(t) \>dt.
$$
Consider first the norm of such a state. Using the reproducing
property of the coherent state, the norm squared becomes a double integral.
\begin{equation}\label{dbint}
\norm{\Phi_{k,\gamma}}^2 = \tfrac{k}{2\pi} \oint\oint e^{ik(\vartheta(t)

\vartheta(s))} e^{k\gbar(s)\gamma(t)  k\gamma(t)^2/2  k\gamma(s)^2/2}
\>\gamma'(t)\>\gamma'(s)\>ds\>dt.
\end{equation}
We can apply stationary phase to the $s$ integration. The equation for the
critical point is
$$
0 =  \vartheta'(s)  i \gamma(t)\gbar'(s) + \tfrac{i}{2}
(\gamma(s)\gbar'(s) + \gbar(s)\gamma'(s)).
$$
Since the curve was horizontal, by (\ref{vart}), this equation reduces to
$$
i(\gamma(s)  \gamma(t))\gbar'(s) = 0.
$$
By the simplicity and regularity of
$\gamma$, the only critical point occurs when $s=t$, and it is
nondegenerate. The second derivative of the phase at the critical
point is $i\gamma'(t)^2$. Applying stationary phase,
$$
\oint e^{ik(\vartheta(t)  \vartheta(s))} e^{k\gbar(s)\gamma(t) 
k\gamma(t)^2/2  k\gamma(s)^2/2}
\>\gamma'(s)\>ds = \sqrt{\tfrac{2\pi}{k}} + k^{1} R(t),
$$
where $R(t)$ is smooth and bounded independently of $k$.
Returning to the norm (\ref{dbint}), we see that
\begin{equation*}
\begin{split}
\norm{\Phi_{k,\gamma}} &= (\tfrac{k}{2\pi})^{1/2}
\oint \gamma'(t)\>dt + O(k^0), \\
&= (\tfrac{k}{2\pi})^{1/2} \times \text{length}(\gamma) + O(k^0).
\end{split}
\end{equation*}
The inner product may be dealt with in a similar way.
We can apply stationary phase to the full integration in
\begin{equation*}
\begin{split}
(\Phi_{k,\gamma_1}, \Phi_{k,\gamma_2})
&= \tfrac{k}{2\pi} \oint\oint e^{ik(\vartheta_2(t) 
\vartheta_1(s))} e^{k\gbar_1(s)\gamma_2(t)  k\gamma_2(t)^2/2 
k\gamma_1(s)^2/2} \\
&\hskip2in \times\gamma_2'(t)\;\gamma_1'(s)\>ds\>dt.
\end{split}
\end{equation*}
Using the condition (\ref{vart}) as above, we see that
critical points occur when $\gamma_1(s) = \gamma_2(t)$.
These will be nondegenerate for transverse intersections.
The result is an asymptotic expression for the inner product
as a sum over intersection points.
The proofs in \cite{BPU1} follow the same basic lines as the
arguments presented here.
The theory of \cite{BG} is used to write the state (locally) as
an oscillatory integral. The main difficulty is to compute the
leading term of this integral.
Then inner products between states are estimated by
stationary phase.
\section{AlmostK\"ahler quantization}
The K\"ahler restriction of Section \ref{kahler} can in fact be dropped.
The Appendix of \cite{BG} shows that one can construct a
``generalized Toeplitz structure,'' namely a projector $\Pi:L^2(P) \to \hil$
with the microlocal properties we described, for any compact contact manifold
$P$. In particular, if $X$ is a compact symplectic manifold with quantizing
circle bundle $P$, then there is a Toeplitz structure on $P$.
The question is: can we define such an $\hil$ in a natural way?
This is also the problem with geometric quantization of $X$. There is no
natural choice of polarization unless $X$ is K\"ahler.
In \cite{BU} a quantization scheme was proposed which at least has the virtue
of generalizing the the K\"ahler case directly. For $X$ K\"ahler one
could think of $\hil_k$ as the kernel of the Laplacian $\dbar^*\dbar$ acting on
sections of $\ltk$. This is quite not the same as the Laplacian on sections
$\Delta_k$ coming from the metric and connection. The two are related by
$\dbar^*\dbar = \Delta_k  nk$.
In the general case $\Delta_k  nk$ will have no kernel. But in \cite{GU} it
was shown that a finite number of eigenfunctions of this shifted Laplacian
stay clustered around 0 as $k\to\infty$. The rest drift to the right like
$O(k)$.
In fact if we define $\hil_k$ to be this collection of lowlying
eigenstates, and set up $\Pi: L^2(P)\to \hil$ as in the
K\"ahler case, this $\Pi$ is a generalized Toeplitz structure.
(This was actually the basis of the proof in \cite{GU}, rather
than the main result.)
Therefore the proofs of semiclassical results in \cite{BMS},
\cite{BPU1}, and \cite{BPU2} apply to this
setting with no alterations.
We have no physical motivation to justify this method of quantizing
by taking low level eigenstates of the Laplacian, and other schemes
are certainly possible. It would be nice to discover
some underlying principle.
More generally, we would like to develop some basic criteria for
quantization theories which will guarantee the kind of good
semiclassical behavior we have seen here.
Such criteria would presumably be microlocal in nature, but hopefully
one could require considerably less structure than the full theory of
\cite{BG}.
\begin{thebibliography}{99}
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Isvestiya} {\bf 8} (1974), 11091163.
\bibitem{B2} F. A. Berezin, General concept of quantization,
{\it Comm. Math. Phys.} {\bf40} (1975), 153174.
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