Analyticity of the Bargman projection
and semiclassical asymptotics
San Vũ Ngo.c
Université de Rennes 1

International Workshop on Geometric Quantization and Applications
CIRM, October 8, 2018

Joint work with Ophélie Rouby and Johannes Sjöstrand
San Vũ Ngo.c, Université de Rennes 1

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Contents

1 Semiclassical analysis, symplectic geometry, and non-selfadjoint

operators

2 Brg-quantization and Bergman kernels

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Contents

1 Semiclassical analysis, symplectic geometry, and non-selfadjoint

operators

2 Brg-quantization and Bergman kernels

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Classical-quantum correspondence
Quantum mechanics:
Quantum state: ψ ∈ H (Hilbert). eg: H = L2 (X).
Quantum observable (Q. Hamiltonian): selfadjoint operator L.
evolution: (Schrödinger equation)~∂t ψ = Lψ.
iitψ(t) = e ~ L ψ(0).
iλtStationary states ψ(t, q) = e ~ u(q): hence Lu = λu.
Classical mechanics:
Phase space: (M, ω) symplectic manifold. eg: M = R2n .
Hamiltonian H ∈ C ∞ (M ).
evolution: Hamiltonian flow
Semiclassical regime: ~ → 0
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Analyticity of the Bargman projectionand semiclassical asymptotics

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Morse Hamiltonian and Reeb graphGeorges Reeb
San Vũ Ngo.c, Université de Rennes 1

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Morse Hamiltonian and Reeb graph(M, ω) a 2-dimensional symplectic manifold (surface). p : M → R
a Morse function. We are interested in the (singular) foliation of
M by level sets of p. The Reeb graph is the set of leaves.

wikimedia

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Morse Hamiltonian and Reeb graph(M, ω) a 2-dimensional symplectic manifold (surface). p : M → R
a Morse function. We are interested in the (singular) foliation of
M by level sets of p. The Reeb graph is the set of leaves.

wikimedia

Question
What is the quantum analogue/footprint1 of the singular foliation?
1

Cf Polterovich’s talk

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Inverse spectral theory for 1D pseudo-differential operators
2
Let P := Opw~ (p~ ) where ph := p + ~p1 + ~ p2 + · · · is a symbol2
on R , elliptic at infinity. Let I ⊂ R be an interval. We assume
M = p−1 (I) is compact (⇒ σ(P ) ∩ I is discrete).

Theorem ([VN,2011])
Suppose that pM is a simple Morse function. Assume that thegraphs of the periods of all trajectories of the hamiltonian flow
defined by pM , as functions of the energy, intersect generically.
Then the knowledge of the spectrum σ(P ) ∩ I + O(~2 ) determines
the symplectic type of (M, ω, p).
Strategy:
detect symplectic invariants
detect singularities
detect topology (Reeb graph)
San Vũ Ngo.c, Université de Rennes 1

use the [Dufour-Molino-Toulet,
1992] classification

Analyticity of the Bargman projectionand semiclassical asymptotics

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Berezin-Toeplitz quantization on Kähler manifolds
Let M be a compact Kähler manifold with a prequantum line
bundle L → M . Let H 0 (X; Lk ) be the space of holomorphic
sections of Lk , for k ≥ 1.
A family of linear operators (Ak : H 0 (X; Lk ) → H 0 (X; Lk ))k≥1 is
a Berezin-Toeplitz operator if there is a smooth symbola : M → R witha ∼ a0 + k −1 a1 + k −2 a2 + · · ·(in the C ∞ topology) such that
Ak = BT (a) := u 7→ Πk (au),where Πk : L2 (X; Lk ) → H 0 (X; Lk )is the orthogonal projection(Bergman projection).

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Berezin-Toeplitz quantization on Kähler manifolds
Let M be a compact Kähler manifold with a prequantum line
bundle L → M . Let H 0 (X; Lk ) be the space of holomorphic
sections of Lk , for k ≥ 1.
A family of linear operators (Ak : H 0 (X; Lk ) → H 0 (X; Lk ))k≥1 is
a Berezin-Toeplitz operator if there is a smooth symbola : M → R with
[Berezin, 1975], [Boutet dea ∼ a0 + k −1 a1 + k −2 a2 + · · ·(in the C ∞ topology) such that
Ak = BT (a) := u 7→ Πk (au),where Πk : L2 (X; Lk ) → H 0 (X; Lk )is the orthogonal projection(Bergman projection).

San Vũ Ngo.c, Université de Rennes 1

Monvel-Guillemin,1981],[Bordemann-MeinrenkenSchlichenmaier,1994],[Borthwick-Paul-Uribe,1998], [Charles, 2003],
etc. . .
and extension to more general symplectic manifolds:[Ma-Marinescu,
2008],[Charles, 2014], etc.

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Inverse spectral theorem for Berezin-Toeplitz operators

Theorem ([Le Floch, 2014])
Let M be a compact, connected Kähler manifold of real dimension
2, with a prequantum line bundle L → M . Let (Ak )k∈N∗ be aself-adjoint Berezin-Toeplitz operator on M whose symbol a is a
simple Morse function (...) Then the knowledge of the spectrum
σ(Ak ) ∩ I + O(k −2 ) determines the symplectic type of (M, ω, a).

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Inverse spectral theorem for Berezin-Toeplitz operators

Theorem ([Le Floch, 2014])
Let M be a compact, connected Kähler manifold of real dimension
2, with a prequantum line bundle L → M . Let (Ak )k∈N∗ be aself-adjoint Berezin-Toeplitz operator on M whose symbol a is a
simple Morse function (...) Then the knowledge of the spectrum
σ(Ak ) ∩ I + O(k −2 ) determines the symplectic type of (M, ω, a).
In particular, we can recover from the spectrum the Reeb graphand its (singular) integral affine structure.

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Inverse spectral theorem for Berezin-Toeplitz operators

Theorem ([Le Floch, 2014])
Let M be a compact, connected Kähler manifold of real dimension
2, with a prequantum line bundle L → M . Let (Ak )k∈N∗ be aself-adjoint Berezin-Toeplitz operator on M whose symbol a is a
simple Morse function (...) Then the knowledge of the spectrum
σ(Ak ) ∩ I + O(k −2 ) determines the symplectic type of (M, ω, a).
In particular, we can recover from the spectrum the Reeb graphand its (singular) integral affine structure.
Remark: recovering the Reeb graph is not obvious (spectrum is
1D).

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Non selfadjoint operators
An idea to “reveal” the topology in the spectrum is to make thespectrum complex. (Non-selfadjoint operators).

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Non selfadjoint operators
An idea to “reveal” the topology in the spectrum is to make thespectrum complex. (Non-selfadjoint operators).
But there is a (huge) price to pay: general non-selfadjoint
operators have very unstable spectrum:
Eg: limit of large Toeplitz matrices, from [Trefethen-Embree, 2005]:

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Bohr-Sommerfeld for analytic non-selfadjoint operators
In order to fight non-selfadjoint instability, we need strong“rigidity” or “integrability” conditions.

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Bohr-Sommerfeld for analytic non-selfadjoint operators
In order to fight non-selfadjoint instability, we need strong“rigidity” or “integrability” conditions. This was recently done foranalytic 1D pseudodifferential operators that are close to
selfadjoint:

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Bohr-Sommerfeld for analytic non-selfadjoint operators
In order to fight non-selfadjoint instability, we need strong“rigidity” or “integrability” conditions. This was recently done foranalytic 1D pseudodifferential operators that are close to
selfadjoint:

Theorem ([Rouby, 2018])
Let P be an analytic pseudodifferential operator on R or S 1 of theform P = P0 + iQ, where P0 is selfadjoint with discrete
spectrum, and Q is P0 -bounded.
Then, near any regular value of the symbol p0 , the spectrum of P
is given by {g(~m; ); m ∈ Z}, where g : C → C is holomorphic
andg ∼ g0 + ~g1 + ~2 g2 + · · ·Moreover, g0 is the inverse of the action variable, andg0 ∼ p0 + ihqi + O(2 ).
[Melin-Sjöstrand, 2003], [Hitrik-Sjöstrand 2004, 2016], [Hitrik-Sjöstrand-VN,
San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Towards Analytic Berezin-Toeplitz calculus
We would like to use this idea for Morse functions on compact
manifolds.

Question
Can you extend Rouby’s theorem to Berezin-Toeplitz quantization?

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Towards Analytic Berezin-Toeplitz calculus
We would like to use this idea for Morse functions on compact
manifolds.

Question
Can you extend Rouby’s theorem to Berezin-Toeplitz quantization?
Then, can you see the Reeb graph?

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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carte postale.qxp_Mise en page 1 13/11/2017 17:35 Page1

carte postale.qxp_Mise en page 1 13/11/2017 17:35 Page2

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Contents

1 Semiclassical analysis, symplectic geometry, and non-selfadjoint

operators

2 Brg-quantization and Bergman kernels

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Zelditch’s question
Recall:BT (a) := u 7→ Πk (au).
All advances in the microlocal calculus of Berezin-Toeplitz
operators rely on the nice semiclassical C ∞ asymptotics of the
Bergman kernel (Schwartz kernel of Πk ), as k → ∞[Fefferman, 1974], [Boutet-Sjöstrand, 1976], [Kashiwara, 1977],...

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Zelditch’s question
Recall:BT (a) := u 7→ Πk (au).
All advances in the microlocal calculus of Berezin-Toeplitz
operators rely on the nice semiclassical C ∞ asymptotics of the
Bergman kernel (Schwartz kernel of Πk ), as k → ∞[Fefferman, 1974], [Boutet-Sjöstrand, 1976], [Kashiwara, 1977],...

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Zelditch’s question
Recall:BT (a) := u 7→ Πk (au).
All advances in the microlocal calculus of Berezin-Toeplitz
operators rely on the nice semiclassical C ∞ asymptotics of the
Bergman kernel (Schwartz kernel of Πk ), as k → ∞[Fefferman, 1974], [Boutet-Sjöstrand, 1976], [Kashiwara, 1977],...
It is not enough for working with analytic symbols (which are usedin the treatment of non-selfadjoint operators).

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Zelditch’s question
Recall:BT (a) := u 7→ Πk (au).
All advances in the microlocal calculus of Berezin-Toeplitz
operators rely on the nice semiclassical C ∞ asymptotics of the
Bergman kernel (Schwartz kernel of Πk ), as k → ∞[Fefferman, 1974], [Boutet-Sjöstrand, 1976], [Kashiwara, 1977],...
It is not enough for working with analytic symbols (which are usedin the treatment of non-selfadjoint operators).

Question (Zelditch, 2014. Also Charles, ...)
Does the Bergman projection Πk admit an asymptotic expansion,
as k → ∞, in the topology of analytic symbols?

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

15/27

Zelditch’s question
Recall:BT (a) := u 7→ Πk (au).
All advances in the microlocal calculus of Berezin-Toeplitz
operators rely on the nice semiclassical C ∞ asymptotics of the
Bergman kernel (Schwartz kernel of Πk ), as k → ∞[Fefferman, 1974], [Boutet-Sjöstrand, 1976], [Kashiwara, 1977],...
It is not enough for working with analytic symbols (which are usedin the treatment of non-selfadjoint operators).

Question (Zelditch, 2014. Also Charles, ...)
Does the Bergman projection Πk admit an asymptotic expansion,
as k → ∞, in the topology of analytic symbols?
[Rouby-Sjöstrand-VN, preprint 2018].

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Zelditch’s question
Recall:BT (a) := u 7→ Πk (au).
All advances in the microlocal calculus of Berezin-Toeplitz
operators rely on the nice semiclassical C ∞ asymptotics of the
Bergman kernel (Schwartz kernel of Πk ), as k → ∞[Fefferman, 1974], [Boutet-Sjöstrand, 1976], [Kashiwara, 1977],...
It is not enough for working with analytic symbols (which are usedin the treatment of non-selfadjoint operators).

Question (Zelditch, 2014. Also Charles, ...)
Does the Bergman projection Πk admit an asymptotic expansion,
as k → ∞, in the topology of analytic symbols?
[Rouby-Sjöstrand-VN, preprint 2018].
Recent simultaneous works by [Hezari-Xu, preprint 2018] and[Deleporte, in preparation].
San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Analytic symbols[Boutet de Monvel-Kree, 1967], [Sjöstrand, 1982]

Definition
A formal classicalanalytic symbol â~ in Ω ⊂ Cn is a formal
P∞
series â~ = j=0 aj ~j , where aj ∈ Hol(Ω) satisfies:∀K b Ω, ∃C > 0, ∀j ≥ 0,

supK |aj | ≤ C j+1 j j .

Definition
We say that a~ ∈ S 0 (Ω) P
is a classical analytic symbol if therejexists a formal cas â~ = ∞j=0 aj ~ , with the asymptotic
expansion: ∀K b Ω, ∃C > 0, ∀N ≥ 0,
sup a~ −K

San Vũ Ngo.c, Université de Rennes 1

N−1
X

aj ~j ≤ ~N C N +1 N N .

j=0
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Holomorphic Hermitian Line Bundles
Let X be a compact complex manifold of complex dimension n, let
L be a holomorphic line bundle over X, equipped with a Hermitian
metric g, giving rise to a metric g k on the tensor product Lk ,k ∈ N∗ .
Assume that g has strictly positive curvature: near any point|s(x)|L = e−Φ(x) , where s is a trivializing section and such that
ω := i∂∂Φ is a Kähler form.
ω induces a volume form ωn on X, and hence a scalar producth·, ·ik on the space of sections of Lk . The orthogonal projection
Πk : L2 (X, Lk ) → H0 (X, Lk )is called the Bergman projection. Its distribution kernel is asmooth section K(·, ·; k) of Fk  Fk∗ :Z
Πk u(x) =K(x, y; k)u(y)ωn (dy).
X
San Vũ Ngo.c, Université de Rennes 1

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Bergman kernels on compact Kähler manifolds
Using trivializing sections s, t near x0 and y0 , we can write
K(x, y; k) = b(x, ȳ; k)sk (x) ⊗ tk (y)∗ .

Theorem (first part: off diagonal)
Assume that the Hermitian metric g is real-analytic.
1

If x0 6= y0 then there exists C > 0 such that, uniformly in aneighborhood Ω0 × V0 of (x0 , y0 ),k|K(x, y; k)|Fk,x ⊗F ∗ ≤ Ce− C .
k,yEquivalently, in local trivializations:ke−k(Φx0 (x)+Φy0 (y)) |b(x, ȳ; k)| = O(e− C ).

San Vũ Ngo.c, Université de Rennes 1

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Bergman kernels on compact Kähler manifolds
Using trivializing sections s, t near x0 and y0 , we can write
K(x, y; k) = b(x, ȳ; k)sk (x) ⊗ tk (y)∗ .

Theorem (first part: off diagonal)
Assume that the Hermitian metric g is real-analytic.
1

If x0 6= y0 then there exists C > 0 such that, uniformly in aneighborhood Ω0 × V0 of (x0 , y0 ),k|K(x, y; k)|Fk,x ⊗F ∗ ≤ Ce− C .
k,yEquivalently, in local trivializations:ke−k(Φx0 (x)+Φy0 (y)) |b(x, ȳ; k)| = O(e− C ).
Mentioned by [Hezari-Lu-Xu, 2018] and [Christ, 2018]. There are many
related results in the literature; see also [Ma-Marinescu, 2015].
San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Bergman kernels on compact Kähler manifolds
Using a trivializing section s near x0 , we can write, for x, y near x0 :K(x, y; k) = b(x, ȳ; k)sk (x) ⊗ sk (y)∗ .

Theorem (second part: near diagonal)
2

For any x0 ∈ X, there exists a neighborhood Ω0 of x0 , and aclassical analytic symbol a on Ω0 × Ω0 , such that, for all(x, y) ∈ Ω0 × Ω0 , for all k ≥ 1,e−k(Φ(x)+Φ(y)) bk (x, ȳ) −k(2k)na(x, ȳ; k −1 )e2kψ(x,ȳ) ≤ Ce− C ,nπ

for some constant C > 0, where ψ is the polarized form of Ψ.
Cf also [Hezari-Lu-Xu, 2018], [Hezari-Xu, preprint 2018].
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Proof strategy
1Construct an approximate Bergman projection that has therequired properties.

2

Prove that the approximate projection is close to the true
projection.

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Proof strategy
1Construct an approximate Bergman projection that has therequired properties.

2

Prove that the approximate projection is close to the true
projection.

Similar strategy has been used e.g. in[Berman-Berndtsson-Sjöstrand, 2008].

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Proof strategy
1Construct an approximate Bergman projection that has therequired properties.

2

Prove that the approximate projection is close to the true
projection.

Similar strategy has been used e.g. in[Berman-Berndtsson-Sjöstrand, 2008].
The “novelty” here is mainly in the way we do item 1. We
simultaneously microlocalize and generalize the problem.

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Proof strategy
1Construct an approximate Bergman projection that has therequired properties.

2

Prove that the approximate projection is close to the true
projection.

Similar strategy has been used e.g. in[Berman-Berndtsson-Sjöstrand, 2008].
The “novelty” here is mainly in the way we do item 1. We
simultaneously microlocalize and generalize the problem.
We consider all operators of the form required by the Theorem
with an analytic symbol a~ (call them Brg-operators).

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Proof strategy
1Construct an approximate Bergman projection that has therequired properties.

2

Prove that the approximate projection is close to the true
projection.

Similar strategy has been used e.g. in[Berman-Berndtsson-Sjöstrand, 2008].
The “novelty” here is mainly in the way we do item 1. We
simultaneously microlocalize and generalize the problem.
We consider all operators of the form required by the Theorem
with an analytic symbol a~ (call them Brg-operators).
And then prove that this quantization is equivalent (by anelliptic analytic FIO) to the complex Weyl quantization.

San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Proof strategy
1Construct an approximate Bergman projection that has therequired properties.

2

Prove that the approximate projection is close to the true
projection.

Similar strategy has been used e.g. in[Berman-Berndtsson-Sjöstrand, 2008].
The “novelty” here is mainly in the way we do item 1. We
simultaneously microlocalize and generalize the problem.
We consider all operators of the form required by the Theorem
with an analytic symbol a~ (call them Brg-operators).
And then prove that this quantization is equivalent (by anelliptic analytic FIO) to the complex Weyl quantization.
Finally, in the Weyl quantization, it is obvious to construct anapproximate Bergman projection.
San Vũ Ngo.c, Université de Rennes 1

Analyticity of the Bargman projectionand semiclassical asymptotics

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Brg quantization
Mimicking the formula:ZΠk u(x) =K(x, y; k)u(y)ωn (dy),X

and working locally in Cn (Φ is a spsh real-analytic function:∂∂Φ ≥ mId, m > 0, and ~ = 1/k), we define OpBrgr (a~ ):

Definition
Let a~ be defined near the anti-diagonal {(x, x̄)} ⊂ Cn × Cn .
ZBrg
k~ (x, y)u(y)L(dy),[Opr (a~ )u](x) :=B(x,r)

where r > 0 andn2(ψ(x,y)−Φ(y))
2
a~ (x, y) det (∂w̃ ∂x ψ) (x, y)k~ (x, y) := (π~)
n e~San Vũ Ngo.c, Université de Rennes 1

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Brg quantization
Mimicking the formula:ZΠk u(x) =K(x, y; k)u(y)ωn (dy),X

and working locally in Cn (Φ is a spsh real-analytic function:∂∂Φ ≥ mId, m > 0, and ~ = 1/k), we define OpBrgr (a~ ):

Definition
Let a~ be defined near the anti-diagonal {(x, x̄)} ⊂ Cn × Cn .
ZBrg
k~ (x, y)u(y)L(dy),[Opr (a~ )u](x) :=B(x,r)

where r > 0 andn2(ψ(x,y)−Φ(y))
2
a~ (x, y) det (∂w̃ ∂x ψ) (x, y)k~ (x, y) := (π~)
n e~Remark: When Φ is quadratic, we have formally ΠΦ = OpBrg∞ (1).
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HΦ spaces
In local coordinates, the holomorphic sections H0 (X, Lk ) are of
the form ũ(x) = u(x)sk (x) for a holomorphic function u such thatZ2kũkk = |u|2 e−2kΦ ωn < +∞,where Φ is spsh: ∂∂Φ ≥ mId, m > 0.

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HΦ spaces
In local coordinates, the holomorphic sections H0 (X, Lk ) are of
the form ũ(x) = u(x)sk (x) for a holomorphic function u such thatZ2kũkk = |u|2 e−2kΦ ωn < +∞,where Φ is spsh: ∂∂Φ ≥ mId, m > 0.
Thus u ∈ HΦ := Hol ∩ L2Φ where L2Φ := L2 (e−2kΦ ).

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HΦ spaces
In local coordinates, the holomorphic sections H0 (X, Lk ) are of
the form ũ(x) = u(x)sk (x) for a holomorphic function u such thatZ2kũkk = |u|2 e−2kΦ ωn < +∞,where Φ is spsh: ∂∂Φ ≥ mId, m > 0.
Thus u ∈ HΦ := Hol ∩ L2Φ where L2Φ := L2 (e−2kΦ ).
If Φ = 12 |z|2 , then HΦ (Cn ) is the usual Bargmann space.

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HΦ spaces
In local coordinates, the holomorphic sections H0 (X, Lk ) are of
the form ũ(x) = u(x)sk (x) for a holomorphic function u such thatZ2kũkk = |u|2 e−2kΦ ωn < +∞,where Φ is spsh: ∂∂Φ ≥ mId, m > 0.
Thus u ∈ HΦ := Hol ∩ L2Φ where L2Φ := L2 (e−2kΦ ).
If Φ = 12 |z|2 , then HΦ (Cn ) is the usual Bargmann space.
Let u ∈ HΦ (Cn ), and ~ := 1/k. We have u = eΦ/~ v, withv ∈ L2 (Cn , |dz|), and the condition ∂u = 0 writes:(~∂z̄ + ∂z̄ Φ)v = 0.
It is a semiclassical equation with characteristic set
Λ̃Φ := {(x, y; ξ, η); (ξ + iη) = 2i∂z̄ Φ(x, y)} ⊂ T ∗ (R2n )San Vũ Ngo.c, Université de Rennes 1

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FBI transforms
Assume here that Φ is quadratic. Then Λ̃Φ is a linear subspace ofT ∗ R2n . If we consider T ∗ R2n ' (T ∗ Rn ) ⊗ C, then Λ̃Φ ' ΛΦ
becomes the image of T ∗ Rn under a complex symplectic map.

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FBI transforms
Assume here that Φ is quadratic. Then Λ̃Φ is a linear subspace ofT ∗ R2n . If we consider T ∗ R2n ' (T ∗ Rn ) ⊗ C, then Λ̃Φ ' ΛΦ
becomes the image of T ∗ Rn under a complex symplectic map.
ΛΦ is the “real phase space” associated with functions inHΦ (Cn ). (i.e., HΦ (Cn ) is the quantization of ΛΦ .)San Vũ Ngo.c, Université de Rennes 1

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FBI transforms
Assume here that Φ is quadratic. Then Λ̃Φ is a linear subspace ofT ∗ R2n . If we consider T ∗ R2n ' (T ∗ Rn ) ⊗ C, then Λ̃Φ ' ΛΦ
becomes the image of T ∗ Rn under a complex symplectic map.
ΛΦ is the “real phase space” associated with functions inHΦ (Cn ). (i.e., HΦ (Cn ) is the quantization of ΛΦ .)At the operator level, we have a unitary transform:
TΦ : L2 (Rn ) → HΦ ⊂ L2Φ (Cn )San Vũ Ngo.c, Université de Rennes 1

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FBI transforms
Assume here that Φ is quadratic. Then Λ̃Φ is a linear subspace ofT ∗ R2n . If we consider T ∗ R2n ' (T ∗ Rn ) ⊗ C, then Λ̃Φ ' ΛΦ
becomes the image of T ∗ Rn under a complex symplectic map.
ΛΦ is the “real phase space” associated with functions inHΦ (Cn ). (i.e., HΦ (Cn ) is the quantization of ΛΦ .)At the operator level, we have a unitary transform:
TΦ : L2 (Rn ) → HΦ ⊂ L2Φ (Cn )Thus, the usual Weyl quantization of T ∗ Rn (acting on
L2 (Rn )) can be transported into a complex Weyl
quantization of ΛΦ (acting on HΦ ). (complex metaplectic
representation)

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FBI transforms
Assume here that Φ is quadratic. Then Λ̃Φ is a linear subspace ofT ∗ R2n . If we consider T ∗ R2n ' (T ∗ Rn ) ⊗ C, then Λ̃Φ ' ΛΦ
becomes the image of T ∗ Rn under a complex symplectic map.
ΛΦ is the “real phase space” associated with functions inHΦ (Cn ). (i.e., HΦ (Cn ) is the quantization of ΛΦ .)At the operator level, we have a unitary transform:
TΦ : L2 (Rn ) → HΦ ⊂ L2Φ (Cn )Thus, the usual Weyl quantization of T ∗ Rn (acting on
L2 (Rn )) can be transported into a complex Weyl
quantization of ΛΦ (acting on HΦ ). (complex metaplectic
representation)[Sjöstrand, 1982] showed how to extend this to the non-linear
setting (Φ not necessarily quadratic), up to exponentiallysmall errors O(e−/~ ).
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Complex Weyl quantization [Sjöstrand, 1982]
Let b~ be a classical analytic symbol defined in a neighborhood of(x0 , θ0 ) ∈ ΛΦ (i.e. θ0 := 2i ∂Φ∂x (x0 )).

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Complex Weyl quantization [Sjöstrand, 1982]
Let b~ be a classical analytic symbol defined in a neighborhood of(x0 , θ0 ) ∈ ΛΦ (i.e. θ0 := 2i ∂Φ∂x (x0 )).

Definition (Complex pseudodifferential operator)Opw~ (b~ )u(x) =1(2π~)n

ZZ
e
Γ(x)

i
(x−y)·θ
~


b~


x+y
, θ u(y)dydθ,2where Γ(x) is a “good contour”:
Γ(x) :=(y, θ) ∈ Ω × Cn ;

θ=

2 ∂Φ
(x) + iR(x − y);i ∂x|x − y| ≤ r .

TheoremIf b is bounded then Opw~ (b~ ) is bounded: HΦ → HΦ−/~mod O(e).
−/~ ).
If b ≡ 1 then Opw~ (b~ ) = Id + O(eSan Vũ Ngo.c, Université de Rennes 1

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A microlocal equivalence
Theorem ([Rouby-Sjöstrand-VN])
Up to exponentially small errors,
1

2

Any Brg-operator OpBrgr (a~ ) can be written as a complexw
pseudodifferential Op~ (b~ ).
Any complex pseudodifferential Opw~ (b~ ) can be written as a
Brg
Brg-operator Opr (a~ );We shall use item 2 to construct an approximate Bergman
projection.

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A microlocal equivalence
Theorem ([Rouby-Sjöstrand-VN])
Up to exponentially small errors,
1

2

Any Brg-operator OpBrgr (a~ ) can be written as a complexw
pseudodifferential Op~ (b~ ).
Any complex pseudodifferential Opw~ (b~ ) can be written as a
Brg
Brg-operator Opr (a~ );We shall use item 2 to construct an approximate Bergman
projection.
Moreover, the map b~ → a~ is an elliptic, analytic FIO.
Therefore, if b~ is an analytic symbol, then a~ is also an analytic
symbol.

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Approximate Bergman projections
Corollary
There exists a classical analytic symbol a~ such that
P := OpBrgr (a~ ) is an approximate Bergman projection:
1

P is selfadjoint;

2

The range of P is HΦ (modulo O(e− /~));

3

P ' Id on HΦ .

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Approximate Bergman projections
Corollary
There exists a classical analytic symbol a~ such that
P := OpBrgr (a~ ) is an approximate Bergman projection:
1

P is selfadjoint;

2

The range of P is HΦ (modulo O(e− /~));

3

P ' Id on HΦ .

Proof: Take the Brg-symbol ah corresponding to the constant
Weyl symbol b~ := 1.


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Approximate Bergman projections
Corollary
There exists a classical analytic symbol a~ such that
P := OpBrgr (a~ ) is an approximate Bergman projection:
1

P is selfadjoint;

2

The range of P is HΦ (modulo O(e− /~));

3

P ' Id on HΦ .

Proof: Take the Brg-symbol ah corresponding to the constant
Weyl symbol b~ := 1.
Finally, since the true Bergman projection ΠΦ is anotherBrg-operator with the same properties, one can show that its
kernel must be exponentially close to the kernel of
P := OpBrgr (a~ ). This proves the main theorem.
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What’s next
A large number of potential applications
Develop symbolic calculus modulo exponentially small
remainders for analytic Berezin-Toeplitz operators(Deleporte?)
Prove Bohr-Sommerfeld spectrum for non-selfadjoint analytic
BT operators.
Non-selfadjoint semitoric operators
Bergman kernel for non compact manifolds?
...

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What’s next
A large number of potential applications
Develop symbolic calculus modulo exponentially small
remainders for analytic Berezin-Toeplitz operators(Deleporte?)
Prove Bohr-Sommerfeld spectrum for non-selfadjoint analytic
BT operators.
Non-selfadjoint semitoric operators
Bergman kernel for non compact manifolds?
...
Thank you for your attention!

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