A.3 A Transformada de Fourier Generalizada
D.0.2 A extens˜ ao A ×
Queremos estender a a¸c˜ao de A : H −→ H para H×. Isto ´e, queremos dar uma a¸c˜ao para A sobre os kets generalizados, i.e. em H×\ H.
A descri¸c˜ao dada na Se¸c˜ao 1.4 faz sentido, mas apenas para A hermitiano. Uma defini¸c˜ao mais geral seria:
hφ| A†|F i = hφ| A×|F i, ∀|F i ∈ H×, ∀ |φi ∈ H, (D.1)
A a¸c˜ao do ket A×|F i ´e definida pela aplica¸c˜ao
A×|F i : H −→ C, |φi 7→ hφ| A†|F i, (D.2)
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H0, 37 Lploc(Ω), 84 H∗, 37 A×, 45 S( Rn), 89 S0 ( Rn), 93 ´Indices de deficiˆencia, 27 Autovalor, 28 Autovetor, 28 Generalizado, 45 Base de Abertos, 20 de Hilbert, 22 Base de Vizinhan¸cas, 29 Bra Limitado, 37 Bras Generalizados, 41 Conjunto Absorvente, 29 Convexo, 29 Equilibrado, 29 Conjunto Enumer´avel, 21 Convergˆencia em S( Rn), 89 Distribui¸c˜oes Temperadas, 93 Enumeravelmente normado, 70 Espa¸co S( Rn), 81 S0 ( Rn), 82 de Sobolev, 97 Localmente Convexo, 29 pr´e-Hilbert, 20 Topol´ogico, 18 Vetorial, 17 Espa¸co de Banach, 20 Espa¸co de Hilbert, 21
Espa¸co de Hilbert Enumer´avel, 33 Espa¸co de Hilbert Equipado, 35 Espa¸co de Schwartz, 89
Topologia, 89 Espa¸co Dual
Conjugado, 37, 38
Espa¸co Normado Enumer´avel, 33 Espa¸co Nuclear, 34
Espa¸co Vetorial Topol´ogico, 29 Espa¸cos Separ´aveis, 22
Espectro
Cont´ınuo, 28
Pontual, 28 Residual, 28 Espectro de um Operador, 28 Estado, 103 Extens˜ao Linear Fechada, 24 Fecho, 21 de um Operador, 24 Fun¸c˜ao Boa, 87
Fun¸c˜ao Delta de Dirac, 88 Fun¸c˜ao Generalizada, 87
Fun¸c˜ao Razoavelmente Boa, 87 Fun¸c˜ao Temperada, 82 Fun¸c˜ao Teste, 83 Funcionais Antilineares, 35 Gerador de Deslocamento, 100 Gr´afido de um Operador, 23 Kets, 42 Norma, 18, 67
Induzida de um Produto Interno, 19 Compat´ıveis, 69 Operador de Hilbert-Schmidt, 70 Densamente definido, 24 EAA, 26 Fechado, 23 Hermitiano, 26 Maximal, 78 Linear, 22 Sim´etrico, 26 Operador Adjunto, 24 Operador C´ıclico, 46 Operador Compacto, 26 Operador Fechavel, 24 Produto Escalar, 67 Produto Interno, 19 Resolvente, 27 Seminorma, 29 Fam´ılia Dirigida, 32 Sequˆencia de Cauchy, 20 Sequˆencia Regular, 87 Sequˆencias Equivalentes, 87 Soma Direta, 67
Subconjunto denso, 21 Suporte
de uma Fun¸c˜ao, 98 Teorema
da representa¸c˜ao de Riesz-Fr´echet, 38 Topologia, 18
Forte, 19
gerada por uma fam´ılia de topologias, 33
Localmente Convexa, 29 Vetorial, 29