430 INDEX
general, 56
linear symbols, 59, 60
on torus, 106–108
standard, 56
symbols
exponentials of linear symbols, 62
exponentials of quadratic symbols,
63
symbols depending on x only, 59
symbols linear in x, 60
Toeplitz, 8, 293, 311–320
Weyl, 4, 6, 56
complex, 312–316
quantum mechanics, 1, 198
Heisenberg picture, 5, 247, 271, 368
quasimode, 152–157, 240–243, 286–288
Quillen’s Theorem, 332
rank, 410
Rank-Nullity Theorem, 418
Rauch–Taylor Theorem, 354
rescaling, 2, 38, 39, 57, 95, 123, 126
standard, 57
Riemannian manifold, 152, 157, 243,
344–345, 365, 378
Riesz Representation Theorem, 101
Riesz–Thorin Theorem, 236
s-density bundles, 342
Schr¨ odinger’s equation, 1, 7
Schur complement formula, 415
Schwartz space S , 28
section, 341
selfadjoint operator, 58, 106, 130, 177,
221, 222, 244, 286, 368, 401, 402,
404–409, 411–413
seminorm, 28, 76, 108, 131, 192, 211,
388
signature of matrix, 36
singular values, 411
Sobolev space, 140, 183, 346, 351, 355
generalized, 7, 182–187, 279
Sogge’s Theorem, 243
spectral clusters, 157, 243
spectrum, 129–132, 177, 286, 287, 357,
403, 405, 408–411
stationary phase, 2, 6, 40–52, 68, 69, 72,
78, 103, 213, 239
higher-dimensional, 46–52
one-dimensional, 40–46
Stirling’s formula, 150, 151, 328
Stone’s Theorem, 222, 409
subadditive function, 265
symbol calculus, 55
symbols, 3, 56, 389
depending only on x, 59
distributional, 58
exponentials of linear symbols, 62
exponentials of quadratic symbols, 63
Kohn–Nirenberg, 7, 206–217, 389
linear, 59, 60
linear in x, 60
symmetric operator, 132, 222, 355, 406
symplectic
form, 342
geometry, 2, 13–26
complex, 299
mapping, 16–20
matrix, 16–17, 252, 262
product σ, 14, 50
tangent
bundle, 341
space, 341
Taylor’s Theorem, 93, 95, 151
tempered
distributions, 35
family of distributions, 187
family of operators, 187, 188
Toeplitz quantization, 293, 311–320
torus, 7, 106–109, 366, 383
trace, 413
integral operators, 413
trace class, 361, 411–413
norm, 411
transform
Bargmann, 292, 306
FBI (Fourier–Bros–Iagolnitzer), 291,
302–311
Fourier, 2
Gabor, 292
Segal–Bargmann, 292
tunneling, 2, 7, 143–148
uncertainty principle, 39–40, 132, 196
unitary
matrix, 253
operators, 85, 100, 126, 172, 176, 220,
222, 228, 246, 251, 256, 259, 260,
263, 368, 404, 408, 409
vector bundles, 340–343
fibers of, 340
sections of, 341
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