Abstract
We develop a new semiclassical calculus in analytic regularity, and apply these techniques to the study of Berezin–Toeplitz quantization in real-analytic regularity. We provide asymptotic formulas for the Bergman projector and Berezin–Toeplitz operators on a compact Kähler manifold. These objects depend on an integer N and we study, in the limit \(N\rightarrow +\infty \), situations in which one can control them up to an error \(O(e^{-cN})\) for some \(c>0\). We develop a calculus of Toeplitz operators with real-analytic symbols, which applies to Kähler manifolds with real-analytic metrics. In particular, we prove that the Bergman kernel is controlled up to \(O(e^{-cN})\) on any real-analytic Kähler manifold as \(N\rightarrow +\infty \). We also prove that Toeplitz operators with analytic symbols can be composed and inverted up to \(O(e^{-cN})\). As an application, we study eigenfunction concentration for Toeplitz operators if both the manifold and the symbol are real-analytic. In this case we prove exponential decay in the classically forbidden region.
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Notes
By this we mean: a real-analytic function \(\kappa \) on \(U\times \Lambda \), where U is a neighbourhood of 0 in \({\widetilde{\Omega }}\), holomorphic in the first variable, such that there exists \(\sigma \) with the same properties, satisfying \(\sigma (\kappa (x,\lambda ),\lambda )=\kappa (\sigma (x,\lambda ),\lambda )=x\) for all \((x,\lambda )\in U\times \Lambda \).
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Acknowledgements
The author thanks L. Charles, N. Anantharaman, S. Vũ Ngọc and J. Sjöstrand for useful discussion.
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This work was supported by Grant ANR-13-BS01-0007-01.
The Wick Rule
The Wick Rule
Here we present a self-contained proof of Proposition 4.4.
It is well known (see [9], Theorem 2) that there exists an invertible formal series a of functions defined on a neighbourhood of the diagonal in \(M\times M\), holomorphic in the first variable and anti-holomorphic in the second variable, which correspond to the Bergman kernel, that is, such that
In Theorem A, we prove that a is in fact an analytic symbol but for the moment, it is sufficient to know that a exists as a formal series.
Let us deform covariant Toeplitz operators by this formal symbol a, into normalised covariant Toeplitz operators of the form \(T_N^{cov}(f* a)\). Here \(*\) denotes the Cauchy product of symbols (Proposition 3.8). Since in this case f and g are simply holomorphic functions one has \(f* a=fa\) and \(g* a=ga\).
We will first prove our claim for this modified quantization: that is, there exists a sequence of bidifferential operators \((C_k)_{k\ge 0}\) acting on functions on a neighbourhood of the diagonal in \(M\times M\), such that, given two such functions f and g, if we let
then
Moreover, \(C_k\) is of order at most k in each of its arguments. Then, we will relate the coefficients \(C_k\) with the coefficients \(B_k\) in the initial claim.
The claim is easier to prove for the coefficients \(C_k\) because normalised covariant Toeplitz quantization follows the Wick rule. Indeed, if the function f, near a point \(x_0\), depends only on the first variable (that is, the restriction of f to the diagonal is, near this point, a holomorphic function on M), then the kernel \(T_N^{cov}(a f)(x,y)\), for x close to \(x_0\), can be written as \(f(x)T_N^{cov}(a)(x,y)=f(x)S_N(x,y)+O(N^{-\infty })\). In particular, for x close to \(x_0\) the Wick rule holds:
since by Remark 4.2 the kernel of \(T_N^{cov}(ag)\) is almost holomorphic in the first variable, up to an \(O(N^{-\infty })\) error. Thus, locally where f depends only on the first variable, there holds
More generally, we wish to compute
where we recall that
Here, we write \((fa)(N)(x,{\overline{y}})\) to indicate that fa is holomorphic in the first variable and anti-holomorphic in the second variable. Similarly, we write \(\Phi _1(x,y,{\overline{w}},{\overline{z}})\) to indicate that \(\Phi _1\) is a function on \(M_x\times {\widetilde{M}}_{y,{\overline{w}}}\times M_{z}\), holomorphic in its two first arguments and anti-holomorphic in the third argument; we integrate over M which is the subset of \({\widetilde{M}}\) such that \({\overline{w}}={\overline{y}}\).
First of all, let us prove a Schur test: operator with kernels of the form
are bounded from \(L^2(M,L^{\otimes N})\) to itself independently on N; in particular, successive integration by parts on \((y,{\overline{y}})\), which will introduce negative powers of N in the symbol, will lead to a control of the operator.
Since for any \((x,z)\in U\) one has \(|\Psi ^N(x,z)|\le e^{-cN{{\,\mathrm{dist}\,}}(x,z)^2}\), then there exists \(C>0\) such that, for any analytic symbol b on \(U\times U\), there holds
In particular, by the Schur test, the operator with the kernel above is bounded independently on N.
As \(\partial _y \Phi _1\) vanishes in a non-degenerate way at \({\overline{w}}={\overline{z}}\), one can write
Thus,
The first term in the right-hand side above is equal to
since \(T_N^{cov}(a)=S_N+O(N^{-\infty })\).
In the second line, which is of order \(N^{-1}\) by a Schur test, derivatives of g of order at most 1 appear. This remainder can be written as
We recover the initial expression, where f has been replaced with either \(F_1\) or \(\partial _yF_1\), and g has potentially been differentiated once. Thus, by induction, the coefficient \(C_k(f,g)\) only differentiates at most k times on g. By duality, \(C_k(f,g)\) only differentiates at most k times on f.
Let us now relate the coefficients \(C_k\) and \(B_k\). Let \(a^{* -1}\) denote the inverse of a for the Cauchy product. One has
so that the coefficients \(B_k\) in the initial claim are recovered as
thus \(B_k\) itself differentiates at most k times on f and at most k times on g.
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Deleporte, A. Toeplitz Operators with Analytic Symbols. J Geom Anal 31, 3915–3967 (2021). https://doi.org/10.1007/s12220-020-00419-w
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DOI: https://doi.org/10.1007/s12220-020-00419-w
Keywords
- Bergman kernel
- Szegö kernel
- Analytic fourier integral operator
- Kähler manifold
- Berezin–Toeplitz operators