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Semiclassical spectral estimates for Toeplitz operators. (English) Zbl 0920.58059

Let \(X\) be a compact Kähler manifold of real dimension \(2n\) and \(L\to X\) a holomorphic Hermitian line bundle such that the curvature of its natural connection is a Kähler form of \(X\). The quantization of \(X\) with Planck’s constant \(1/k\) is the space \(H^0(x,L^{\otimes k})\), and the Toeplitz quantization of a classical Hamiltonian function \(H\in C^\infty(X)\) is \(S_k= \pi_k M_H\pi_k\), where \(\pi_k: L^2(X,L^{\otimes k})\to H^0(X, L^{\otimes k})\) is the orthogonal projector and \(M_H\) is the operator of multiplication by \(H\). Let \(P\subset L^*\) be the unit circle bundle in the dual of \(L\). By the condition that the curvature of \(L\) be the Kähler form, \(P\) is strictly pseudoconvex and the contact form \(\alpha\) is the connection form on \(P\).
Let \(\partial_\theta\) denote the contact vector field on \(P\), uniquely determined by \(\alpha\) (\(\partial_\theta\) generates a free \(S^1\) \((=\mathbb{R}/2\pi \mathbb{Z})\) action of automorphisms of \(P\)). The operator \(\partial_\theta\) determines the decomposition of the Hardy space of \(P\) into eigenspaces of \(\partial_\theta: {\mathcal H}=\oplus{\mathcal H}_k\). In this decomposition one can naturally make the identification: \({\mathcal H}_k= H^0(X, L^{\otimes k})\). Under this identification \(S_k\) gets identified with the operator on \({\mathcal H}_k\) with eigenvalues and eigenfunctions \((E_i^k, \psi_i^k)\).
Let \(\phi\) be Hamiltonian flow of \(H\) on \(X\) and \(\pi:P\to X\) the obvious projection.
The first main result (Theorem 1.1) is concerned with the integral kernel of \(\varphi(k(T_k- E))\), where \(T\) is the Toeplitz operator on \({\mathcal H}\), \(T_k= T|{\mathcal H}_k\) and \(\varphi\) is a smooth test function with compactly supported Fourier transform, and the second one (Theorem 1.2) concerns the trace of \(\varphi (k(T_k-E))\).
Theorem 1.1. Let \(p_1,p_2\in P\), \(x_i= \pi(p_i)\) and in case \(p_1=p_2\) the point \(x_1= x_2\) is not fixed of \(\phi\). Then:
(1) if \(H(x_1)= H(x_2)= E\) and \(x_1,x_2\) lie on the same orbit of \(\phi\), then the sums \((*)\) \(\sum_{i=0}^\infty \varphi(k(T_k- E))\psi_i^k (p_1) \overline{\psi_i^k (p_2)}\) admit an asymptotic expansion as \(k\to\infty\) of the form \[ \sum_r \sum_{j=0}^\infty C_{r,j} \widehat{\varphi}(r) e^{ikt_r} k^{n-{1/2}-y}, \] where \(r\in \mathbb{R}\) and \(\phi_r(x_2)= x_1\).
(2) If \(\{x_1:x_2\} \not\subset H^{-1}(E)\) or \(x_1,x_2\) are not connected by any \(\phi\)-trajectory, then \((*)\) decreases rapidly in \(k\).

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J40 Pseudodifferential and Fourier integral operators on manifolds
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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[1] [1] , Une classe caractéristique intervenant dans les conditions de quantification, in V. P.MASLOV, Théorie des perturbations et Méthodes asymptotiques, Dunod, Paris (1972) 341-361.
[2] [2] , General concept of quantization, Comm. Math. Phys., 40 (1975), 153-174. · Zbl 1272.53082
[3] [3] and , The quantized Baker’s transformation, Annals of Physics, 180 (1989), 1-31. · Zbl 0664.58045
[4] [4] , , and , Toeplitz quantization of Kähler manifolds and gl(N), N → ∞ limits, Comm. Math. Phys., 165 (1994), 281-296. · Zbl 0813.58026
[5] [5] , , and , Legendrian distributions and non-vanishing of Poincaré series, Invent. Math., 122 (1995), 359-402. · Zbl 0859.58015
[6] [6] , On the index of Toeplitz operators of several complex variables, Invent. Math., 50 (1979), 249-272. · Zbl 0398.47018
[7] [7] , Hypoelliptic operators with double characteristics and related pseudodifferentiel operators, Comm. Pure Appl. Math., 27 (1974), 585-639. · Zbl 0294.35020
[8] [8] and , The spectral theory of Toeplitz operators. Annals of Mathematics Studies No. 99, Princeton University Press, Princeton, New Jersey (1981). · Zbl 0469.47021
[9] [9] and , Sur la singularité des noyaux de Bergmann et de Szego, Astérisque, 34-35 (1976), 123-164. · Zbl 0344.32010
[10] [10] , , and , Quantization of Kähler manifolds. I: geometric interpretation of Berezin’s quantization, J. Geom. Phys. 7 (1990) 45-62; Quantization of Kähler manifolds. II, Trans. Amer. Math. Soc., 337 (1993) 73-98; Quantization of Kähler manifolds. III, preprint (1993). · Zbl 0719.53044
[11] [11] , and , Stochastic properties of the quantum Arnol’d cat in the classical limit, Comm. Math. Phys., 167 (1995), 471-509. · Zbl 0822.58022
[12] [Ga] , Projections of semi-analytic sets, Funct. Anal. Appl., 2 ( · Zbl 0179.08503
[13] [13] and , The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math., 29 (1975), 39-79. · Zbl 0307.35071
[14] [14] , Harmonic Analysis in Phase Space, Annals of Mathematics Studies 122, Princeton University Press, Princeton N.J. 1989. · Zbl 0682.43001
[15] [15] and , Quantum intrinsically degenerate and classical secular perturbation theory, preprint.
[16] [16] , Symplectic spinors and partial differential equations. Coll. Inst. CNRS 237, Géométrie Symplectique et Physique Mathématique, 217-252. · Zbl 0341.58014
[17] [17] and , Geometric quantization and multiplicities of group representations, Invent. Math., 67 (1982), 515-538. · Zbl 0503.58018
[18] [18] and , Circular symmetry and the trace formula, Invent. Math., 96 (1989), 385-423. · Zbl 0686.58040
[19] [LR2] , , Théorème de Gabrielov et fonctions log-exp-algébriques, preprint (1996). · Zbl 1194.81107
[20] [20] , The analysis of linear partial differential operators I-IV, Springer-Verlag, 1983-1985. · Zbl 0612.35001
[21] [21] and , The semi-classical trace formula and propagation of wave packets, J. Funct. Analysis, 132, No.1 (1995), 192-249. · Zbl 0837.35106
[22] [22] and , On the pointwise behavior of semi-classical measures, Comm. Math. Phys., 175 (1996), 229-258. · Zbl 0853.47038
[23] [23] and , Weighted Weyl estimates near an elliptic trajectory, Revista Matemática Iberoamericana, 14 (1998), 145-165. · Zbl 0923.58055
[24] [24] , Autour de l’approximation semi-classique, Birkhauser 1987. · Zbl 0621.35001
[25] [25] and , Semiclassical spectra of gauge fields, J. Funct. Anal., 110 (1992), 1-46. · Zbl 0772.58066
[26] [26] , Large N limits as classical mechanics, Rev. Mod. Phys., 54 (1982), 407-435.
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