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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