428 INDEX quantum, 5, 106, 219–226, 245–251, 264–271, 409 Egorov’s Theorem, 8, 245–251, 278 for long times, 8, 264–271 weak, 368–370 Ehrenfest time, 8, 264–271 eigenfunctions, 7, 119–137, 143, 152 basis of, 129, 157, 357 clusters of, 157, 243 concentration in phase space, 125 equidistribution of, 365, 378, 379 exponential decay estimates, 143 for harmonic oscillator, 120–124, 127 for Laplace–Beltrami operator, 357, 378 for pseudodifferential operator, 187 on manifolds, 356 order of vanishing, 152 regularity, 354 eigenvalues, 48, 120, 123, 286, 287, 357, 362, 363, 370, 404, 405, 411, 412 and trace, 413 counting, 410 for harmonic oscillator, 120–124 for Laplace–Beltrami operator, 157, 243, 357, 362 for Schr¨ odinger’s equation, 1, 7, 119, 286, 357 minimax formulas for, 409, 410 of matrix, 36 of operator, 403 elliptic estimates, 140 symbol, 91, 133, 144, 146, 156, 222, 242, 260, 277, 282, 288, 289 energy decay, 114–117 surface, 366 wave equation, 109 ergodicity, 8 classical, 366–368 quantum, 365–379 essential support, 192–194 essentially selfadjoint operator, 406 estimates Hh, 2 140 Hh, k 149 Agmon–Lithner, 142 Carleman, 7, 146–148 notation for, 388 Schauder, 7, 158–167 Strichartz, 8, 235–240 exponential map, 14 FBI transform, 291, 302–311 flow map, 14 forbidden region, 139, 141 Fourier decomposition, 65 integral operator, 8, 199, 228, 244, 245 Fourier transform, 2, 27–40 exponential of imaginary quadratic form, 36 exponential of real quadratic form, 28 on S , 28 on S , 36 semiclassical, 38–40 Fredholm operator, 416 theory, 415–419 functional calculus, 137, 354, 357–361, 370 generalized Sobolev space Hh(m), 182–187 definition, 183 dual space of, 184 examples, 183 pseudodifferential operators and, 185, 187 geodesic flow, 365, 378, 379 graph, 405 twisted, 25, 262, 383 Grushin problems, 415–417 half-density, 197–206, 234–235, 345 Hamilton–Jacobi equation, 7, 228, 231–233, 238, 260 harmonic oscillator, 119–124, 126, 135 Weyl’s Law for, 123 heat equation, 285 Helffer–Sj¨ ostrand formula, 358, 361, 363 Helmholtz’s equation, 274 Hermite polynomials, 121 hypoellipticity condition, 144, 145 estimate, 144 Implicit Function Theorem, 19, 25, 156, 366 index of Fredholm operator, 416 inequality Fefferman–Phong, 93, 216

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