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Quantum transition amplitudes for ergodic and for completely integrable systems. (English) Zbl 0721.58051
It is well known that the asymptotic distribution of eigenvalues, \(\lambda_ j\), of the Laplacian on a compact Riemannian manifold, M, reflects geometrical properties of the manifold. In particular the uniformity of the distribution depends upon the extent to which the geodesics on M fail to be closed. The purpose of this paper is to draw attention to the asymptotic (in j) behaviour of the corresponding eigenfunctions \(\phi_ j\) and the dynamics of the geodesic flows and particularly to discuss the cases where the flow is either mixing or completely integrable. To quantify the asymptotics of the \(\phi_ j\) one looks at \((A\phi_ j,\phi_ i)\) where A is a zeroth order pseudodifferential operator on \(L^ 2(M)\). (These are the “quantum transitions amplitudes” of the title.) This is a linear functional of the symbol, \(\sigma_ A\in C^{\infty}(T^*M)\), and one wishes to know the weak limits of subsequences of \(\{d\Phi_{i,j}\}\) where \(d\Phi_{i,j}\in {\mathcal D}'(T^*M)\) is defined by \(<\sigma_ A,d\Phi_{i,j}>=(A\phi_ i,\phi_ j)\). A result of A. I. Shnirel’man [Usp. Mat. Nauk 29, No.6(180), 181-182 (1974; Zbl 0324.58020)] states that \(\{d\Phi_{i,i}\}\) \((i=j)\) tends to Liouville measure if the geodesic flow is ergodic. The author considers off- diagonal elements (i\(\neq j)\). Thus if the flow is ergodic and a positive proportion of eigenvalues are multiple then, up to a sparse subsequence, \(d\Phi_{i_ r,j_ r}\to 0\) when \(i_ r\neq j_ r\) but \(\lambda_{i_ r}=\lambda_{j_ r}\). This also holds when the flow is mixing. For completely integrable flows \(T^*M\) is foliated by invariant tori and certain subsequences of \(\{d\Phi_{i,j}\}\) tend to \(\delta\)- function on these tori. More precisely the asymptotic limit is given by an orbital Fourier coefficient on the torus.

MSC:
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37A99 Ergodic theory
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[1] de Verdiere, Y.Colin, Spectre conjoint d’opérateurs pseudo-différentiels qui commutent. II. le cas intégrable, Math. Z., 171, 51-73, (1980) · Zbl 0478.35073
[2] de Verdiere, Y.Colin, Quasi-modes sur LES variétés riemannienes compactes, Invent. math., 43, 15-52, (1977) · Zbl 0449.53040
[3] de Verdiere, Y.Colin, Ergodicité et fonctions propres du laplacien, Comm. math. phys., 102, 497-502, (1985) · Zbl 0592.58050
[4] Duistermaat, H; Guillemin, V, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. math., 29, 39-79, (1975) · Zbl 0307.35071
[5] Guillemin, V, Some classical theorems in spectral theory revisited, () · Zbl 0452.35093
[6] Guillemin, V, Lectures on spectral theory of elliptic operators, Duke math. J., 44, 485-517, (1977) · Zbl 0463.58024
[7] Guillemin, V; Sternberg, S, Homogeneous quantization and multiplicities of group representations, J. funct. anal., 47, 344-380, (1982) · Zbl 0733.58021
[8] Guillemin, V; Sternberg, S, The metaplectic representation, Weyl operators, and spectral theory, J. funct. anal., 42, 128-225, (1981) · Zbl 0469.58017
[9] Hormander, L, The Weyl calculus of pseudo-differential operators, Comm. pure appl. math., 32, 359-443, (1979) · Zbl 0388.47032
[10] Pedersen, K, Ergodic theory, (1983), Cambridge Univ. Press London/New York
[11] Snirelman, A.I, Ergodic properties of eigenfunctions, Uspekhi mat. nauk, 29, 181-182, (1974) · Zbl 0324.58020
[12] Sigmund, K, On the space of in variant measures for hyperbolic flows, Ann. of math., 94, 31-37, (1972) · Zbl 0242.28014
[13] Uribe, A, Eigenfunctions in the completely integrable case, (1985), preprint
[14] Walters, P, An introduction to ergodic theory, (1982), Springer-Verlag New York · Zbl 0475.28009
[15] Widom, H, Eigenvalue distribution theorems for certain homogeneous spaces, J. funct. anal., 32, 139-147, (1979) · Zbl 0414.43010
[16] Zelditch, S, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke math. J., 55, 919-941, (1987) · Zbl 0643.58029
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