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Toeplitz Operators with Analytic Symbols

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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

  1. 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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Correspondence to Alix Deleporte.

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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

$$\begin{aligned} T_N^{cov}(a)=S_N+O(N^{-\infty }). \end{aligned}$$

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

$$\begin{aligned} f\sharp g=\sum _{k=0}^{+\infty }N^{-k}C_k(f,g)+O(N^{-\infty }), \end{aligned}$$

then

$$\begin{aligned} T_N^{cov}((f\sharp g)* a)=T_N^{cov}(fa)T_N^{cov}(ga)+O(N^{-\infty }). \end{aligned}$$

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:

$$\begin{aligned} T_N^{cov}(af)T_N^{cov}(ag)(x,y)=T_N^{cov}(afg)(x,y)+O(N^{-\infty }), \end{aligned}$$

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

$$\begin{aligned} \forall k\ge 1,\,C_k(f,g)=0. \end{aligned}$$

More generally, we wish to compute

$$\begin{aligned} N^{2d}\Psi ^N(x,z){\int }_{M}\exp (N\Phi _1(x,y,{\overline{y}},{\overline{z}})) (fa)(N)(x,{\overline{y}})(ga)(N)(y,{\overline{z}})\mathrm {d}y, \end{aligned}$$

where we recall that

$$\begin{aligned} \Phi _1(x,y,{\overline{w}},{\overline{z}})=-2{\widetilde{\phi }}(x,{\overline{w}}) +2{\widetilde{\phi }}(y,{\overline{w}})-2{\widetilde{\phi }}(y,{\overline{z}})+ 2{\widetilde{\phi }}(x,{\overline{z}}). \end{aligned}$$

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

$$\begin{aligned} (x,z)\mapsto N^{2d}{\int }_M\exp (N\Phi _1(x,y,{\overline{y}}, {\overline{z}}))b(x,y,{\overline{y}},z)\mathrm {d}y \end{aligned}$$

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

$$\begin{aligned}&N^{2d}\sup _{x}{\int }_{M}\left| \Psi ^N(x,z){\int }_M\exp (N\Phi _1(x,y,{\overline{y}},{\overline{z}})) b(N)(x,y,{\overline{y}},z)\mathrm {d}y\right| \mathrm {d}z \\&\quad \le N^{2d}\sup _{U\times U}|b(N)|\sup _x{\int }_M{\int }_M|\Psi ^N(x,y)||\Psi ^N(y,z)|\mathrm {d}y \mathrm {d}z\\&\quad \le \sup _{U\times U}|b(N)|N^{2d}\sup _x{\int }_{M\times M}e^{-Nc{{\,\mathrm{dist}\,}}(x,y)^2-Nc{{\,\mathrm{dist}\,}}(y,z)^2}\mathrm {d}y \mathrm {d}z\\&\quad \le C\sup _{U\times U}|b(N)|. \end{aligned}$$

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

$$\begin{aligned} f(x,{\overline{w}})=f(x,{\overline{z}})-\partial _y\Phi _1\cdot F_1(x,{\overline{z}},y,{\overline{w}}). \end{aligned}$$

Thus,

$$\begin{aligned}&N^{2d}\Psi ^N(x,z){\int }_{M}e^{N\Phi _1(x,y,{\overline{y}},{\overline{z}})}(fa)(N)(x,{\overline{y}}) (ga)(N)(y,{\overline{z}})\mathrm {d}y \\&\quad = N^{2d}\Psi ^N(x,z)f(x,{\overline{z}}) \left( {\int }_Me^{N\Phi _1(x,y,{\overline{y}},{\overline{z}})}a(N)(x,{\overline{y}})(ga)(N)(y,{\overline{z}})\mathrm {d}y\right. \\&\qquad \left. +N^{-1} {\int }_Me^{N\Phi _1(x,y,{\overline{y}},{\overline{z}})}a(N)(x,{\overline{y}})\partial _M \left[ F_1(x,{\overline{z}},y,{\overline{y}})(g a)(N)(y,{\overline{z}})\right] \mathrm {d}y\right) . \end{aligned}$$

The first term in the right-hand side above is equal to

$$\begin{aligned} f(x,{\overline{z}}){\int }_M T_N^{cov}(a)(x,{\overline{y}})T_N^{cov}(g a)(y,{\overline{z}})\mathrm {d}y=f(x,{\overline{z}})T_N^{cov}(g a)(x,{\overline{z}})+O(N^{-\infty }), \end{aligned}$$

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

$$\begin{aligned}&N^{2d-1}\Psi ^N(x,{z}){\int }_Me^{N\Phi _1(x,y,{\overline{y}},{\overline{z}})}a(N)(x,{\overline{y}}) \left[ \partial _yF_1(x,{\overline{z}},y,{\overline{y}})\right] (g a)(N)(y,{\overline{z}})\mathrm {d}y\\&\quad + N^{2d-1}\Psi ^N(x,{z}){\int }_Me^{N\Phi _1(x,y,{\overline{y}},{\overline{z}})}a(N)(x,{\overline{y}}) F_1(x,{\overline{z}},y,{\overline{y}})[\partial _y(ga)(N)(y,{\overline{z}})\mathrm {d}y. \end{aligned}$$

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

$$\begin{aligned} T_N^{cov}(f)T_N^{cov}(g)&=T_N^{cov}((f a^{* -1})* a)T_N^{cov}((g a^{* -1})* a)+O(N^{-\infty })\\&=T_N^{cov}((C_k(f,g))_{k\ge 0}* a)+O(N^{-\infty }), \end{aligned}$$

so that the coefficients \(B_k\) in the initial claim are recovered as

$$\begin{aligned} B_k(f,g)=\sum _{j+l+m\le k}a_jC_{k-j-l-m}(fa^{* -1}_l,ga^{* -1}_m), \end{aligned}$$

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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