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