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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