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1 | (23) |
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General introductory comments |
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1 | (7) |
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1 | (3) |
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Feynman's operational calculus |
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4 | (1) |
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Feynman's operational calculus via the Feynman and Wiener integrals |
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5 | (2) |
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Feynman's operational calculus and evolution equations |
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7 | (1) |
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Further work on or related to the Feynman integral:
Chapter 20 |
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8 | (1) |
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Recurring themes and their connections with the Feynman integral and Feynman's operational calculus |
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8 | (9) |
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Product formulas and applications to the Feynman integral |
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8 | (2) |
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Feynman--Kac formula: Analytic continuation in time and mass |
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10 | (2) |
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The role of operator theory |
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12 | (1) |
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Connections between the Feynman--Kac and Trotter product formulas |
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13 | (1) |
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13 | (2) |
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Functions of noncommuting operators |
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15 | (1) |
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Time-ordered perturbation series |
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15 | (1) |
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16 | (1) |
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Relationship with the motivating physical theories: background and quantum-mechanical models |
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17 | (7) |
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17 | (1) |
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Highly singular potentials |
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18 | (1) |
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Time-dependent potentials |
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19 | (1) |
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Phenomenological models: complex and nonlocal potentials |
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19 | (2) |
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Prerequisites, new material, and organization of the book |
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21 | (3) |
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The physical phenomenon of Brownian motion |
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24 | (7) |
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A brief historical sketch |
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24 | (4) |
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Einstein's probabilistic formula |
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28 | (3) |
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31 | (31) |
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There is no reasonable translation invariant measure on Wiener space |
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32 | (2) |
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Construction of Wiener measure |
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34 | (8) |
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Wiener's integration formula and applications |
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42 | (9) |
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43 | (2) |
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45 | (6) |
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Axiomatic description of the Wiener process |
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51 | (1) |
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Nondifferentiability of Wiener paths |
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51 | (6) |
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d-dimensional Wiener measure and Wiener process |
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57 | (1) |
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Appendix: Converse measurability results |
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57 | (3) |
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Appendix: B(X x Y) = B(X) B(Y) |
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60 | (2) |
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Scaling in Wiener space and the analytic Feynman integral |
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62 | (27) |
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Quadratic variation of Wiener paths |
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63 | (4) |
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Scale change in Wiener space |
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67 | (7) |
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74 | (3) |
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Scale-invariant measurable functions |
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77 | (2) |
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The scalar-valued analytic Feynman integral |
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79 | (3) |
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The nonexistence of Feynman's ``measure'' |
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82 | (3) |
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Appendix: Some useful Gaussian-type integrals |
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85 | (2) |
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Appendix: Proof of formula (4.2.3a) |
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87 | (2) |
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Stochastic processes and the Wiener process |
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89 | (5) |
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Stochastic processes and probability measures on function spaces |
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89 | (1) |
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The Kolmogorov consistency theorem |
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90 | (2) |
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Two realizations of the Wiener process |
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92 | (2) |
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Quantum dynamics and the Schrodinger equation |
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94 | (5) |
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Hamiltonian approach to quantum dynamics |
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94 | (1) |
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Transition amplitudes and measurement |
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95 | (1) |
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The Heisenberg uncertainty principle |
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96 | (1) |
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Hamiltonian for a system of particles |
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97 | (2) |
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The Feynman integral: heuristic ideas and mathematical difficulties |
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99 | (22) |
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99 | (2) |
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101 | (5) |
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Connections with classical mechanics: The method of stationary phase |
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105 | (1) |
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Heuristic derivation of the Schrodinger equation |
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106 | (3) |
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Feynman's approximation formula |
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109 | (2) |
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Nelson's approach via the Trotter product formula |
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111 | (4) |
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The Trotter product formula |
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114 | (1) |
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The approach via analytic continuation |
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115 | (6) |
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Semigroups of operators: an informal introduction |
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121 | (6) |
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Linear semigroups of operators |
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127 | (25) |
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127 | (7) |
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129 | (1) |
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130 | (1) |
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Closed unbounded operators |
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130 | (4) |
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Examples of semigroups and their generators |
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134 | (4) |
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The translation semigroup |
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134 | (1) |
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135 | (2) |
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137 | (1) |
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138 | (2) |
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140 | (3) |
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The Hille--Yosida theorem |
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140 | (1) |
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Dissipative operators and the Lumer--Phillips theorem |
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141 | (2) |
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Uniformly continuous and weakly continuous semigroups |
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143 | (1) |
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Self-adjoint operators, unitary groups and Stone's theorem |
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144 | (3) |
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147 | (5) |
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Unbounded self-adjoint operators and quadratic forms |
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152 | (45) |
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Spectral theorem for unbounded self-adjoint operators |
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153 | (8) |
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153 | (2) |
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Three useful forms of the spectral theorem |
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155 | (6) |
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Applications of the spectral theorem |
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161 | (15) |
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161 | (2) |
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The heat semigroup and unitary group |
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163 | (8) |
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Standard cores for the free Hamiltonian |
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171 | (3) |
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174 | (2) |
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Representation theorems for unbounded quadratic forms |
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176 | (17) |
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Basic definitions and properties |
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176 | (7) |
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Representation theorems for quadratic forms |
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183 | (7) |
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The form sum of operators |
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190 | (3) |
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Conditions on the potential V for H0-form boundedness |
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193 | (4) |
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Product formulas with applications to the Feynman integral |
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197 | (75) |
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Trotter and Chernoff product formulas |
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198 | (6) |
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Product formula for unitary groups |
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203 | (1) |
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Feynman integral via the Trotter product formula |
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204 | (16) |
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Criteria for essential self-adjointness of positive operators |
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204 | (1) |
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A brief outline of distribution theory |
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205 | (1) |
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Kato's distributional inequality |
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206 | (4) |
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Essential self-adjointness of the Hamiltonian H = H0 + V |
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210 | (5) |
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Conditions on the potential V for H0-operator boundedness |
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215 | (2) |
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Feynman integral via the Trotter product formula for unitary groups |
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217 | (3) |
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Product formula for imaginary resolvents |
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220 | (12) |
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Hypotheses and statement of the main result |
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220 | (2) |
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Proof of the product formula |
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222 | (7) |
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Consequences, extensions and open problems |
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229 | (3) |
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Application to the modified Feynman integral |
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232 | (13) |
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Modified Feynman integral and Schrodinger equation with singular potential |
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233 | (8) |
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Extensions: Riemannian manifolds and magnetic vector potentials |
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241 | (4) |
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Dominated convergence theorem for the modified Feynman integral |
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245 | (14) |
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247 | (1) |
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Perturbation of form sums of self-adjoint operators |
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248 | (4) |
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Application to a general dominated convergence theorem for Feynman integrals |
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252 | (7) |
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The modified Feynman integral for complex potentials |
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259 | (7) |
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Product formula for imaginary resolvents of normal operators |
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260 | (3) |
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Application to dissipative quantum systems |
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263 | (3) |
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Appendix: Extended Vitali's theorem with application to unitary groups |
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266 | (6) |
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Extension of Vitali's theorem for sequences for analytic functions |
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266 | (3) |
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Analytic continuation and product formula for unitary groups |
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269 | (3) |
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272 | (21) |
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The Feynman-Kac formula, the heat equation and the Wiener integral |
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273 | (3) |
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Proof of the Feynman--Kac formula |
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276 | (14) |
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276 | (6) |
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Monotone convergence theorems for forms and integrals |
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282 | (2) |
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284 | (6) |
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290 | (3) |
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Analytic-in-time or-mass operator-valued Feynman integrals |
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293 | (81) |
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293 | (5) |
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The analytic-in-time operator-valued Feynman integral |
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298 | (2) |
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300 | (3) |
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The Feynman integrals compared with one another and with the unitary group. Application to stability theorems |
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303 | (5) |
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The analytic-in-mass operator-valued Feynman integral |
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308 | (23) |
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Definition of the analytic-in-mass operator-valued Feynman integral |
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309 | (3) |
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312 | (9) |
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Haugsby's result for time-dependent, complex-valued potentials |
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321 | (2) |
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Further extensions via a product formula for semigroups |
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323 | (8) |
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The analytic-in-mass modified Feynman integral |
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331 | (27) |
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Existence of the analytic-in-mass modified Feynman integral |
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334 | (3) |
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Product formula for resolvents: The case of imaginary mass |
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337 | (8) |
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Comparison with other analytic-in-mass Feynman integrals |
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345 | (1) |
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Highly singular central potentials---the attractive inverse-square potential |
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346 | (12) |
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The analytic-in-time operator-valued Feynman integral via additive functionals of Brownian motion |
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358 | (16) |
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359 | (1) |
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The parallel with Section 13.3 |
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359 | (2) |
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Generalized signed measures |
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361 | (1) |
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The generalized Kato class |
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361 | (1) |
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362 | (1) |
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363 | (1) |
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Positive continuous additive functionals of Brownian motion |
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364 | (1) |
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The relationship between smooth measures and PCAFs |
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365 | (2) |
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The analytic-in-time operator-valued Feynman integral exists for μ = μ+ - μ_ S -- GKd |
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367 | (1) |
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368 | (6) |
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Feynman's operational calculus for noncommuting operators: an introduction |
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374 | (30) |
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375 | (1) |
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The rules for Feynman's operational calculus |
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376 | (7) |
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Feynman's time-ordering convention |
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377 | (1) |
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Feynman's heuristic rules |
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377 | (1) |
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378 | (5) |
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Time-ordered Perturbation series |
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383 | (11) |
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Perturbation series via Feynman's operational calculus |
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383 | (6) |
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Perturbation series via a path integral |
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389 | (4) |
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The origins of Feynman's operational calculus rigorous |
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393 | (1) |
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Making Feynman's operational calculus rigorous |
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394 | (3) |
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394 | (1) |
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Well-defined and useful formulas arrived at via Feynman's heuristic rules |
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395 | (1) |
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A general theory of Feynman's operational calculus with computations which are rigorous at each stage |
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396 | (1) |
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Feynman's operational calculus via Wiener and Feynman integrals: Comments on
Chapter 15--18 |
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397 | (7) |
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Generalized Dyson series, the Feynman integral and Feynman's operational calculus |
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404 | (58) |
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404 | (3) |
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The analytic operator-valued Feynman integral |
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407 | (9) |
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407 | (3) |
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The analytic (in mass) operator-valued Feynman integral Ktλ(·) |
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410 | (3) |
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413 | (3) |
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A simple generalized Dyson series (η = μ + ωδτ) |
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416 | (10) |
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The classical Dyson series |
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424 | (2) |
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Generalized Dyson series: The general case |
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426 | (8) |
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Disentangling via perturbation expansions: Examples |
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434 | (12) |
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A single measure and potential |
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435 | (7) |
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Several measures and potentials |
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442 | (4) |
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Generalized Feynman diagrams |
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446 | (5) |
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Commutative Banach algebras of functionals: The disentangling algebras |
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451 | (11) |
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The disentangling algebras At |
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452 | (3) |
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The time-reversal map on At and the natural physical ordering |
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455 | (4) |
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Connections with Feynman's operational calculus |
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459 | (3) |
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462 | (15) |
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Stability in the potentials |
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462 | (2) |
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Stability in the measures |
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464 | (13) |
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The Feynman--Kac formula with a Lebesgue--Stieltjes measure and Feynman's operational calculus |
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477 | (53) |
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477 | (3) |
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478 | (2) |
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The Feynman--Kac formula with a Lebesgue--Stieltjes measure: Finitely supported discrete part v |
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480 | (6) |
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Integral equation (Integrated form of the evolution equation) |
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480 | (1) |
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Differential equation (differential form of the evolution equation) |
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481 | (1) |
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Discontinuities (in time) of the solution |
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482 | (1) |
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Propagator and explicit solution |
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483 | (3) |
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Derivation of the integral equation in a simple case (η = μ + ωδτ) |
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486 | (10) |
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Sketch of the proof when v is finitely supported |
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495 | (1) |
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Discontinuities of the solution to the evolution equation |
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496 | (3) |
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496 | (1) |
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Differential equation and change of initial condition |
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497 | (2) |
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Explicit solution and physical interpretations |
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499 | (8) |
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Continuous measure: Uniqueness of the solution |
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500 | (1) |
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Measure with finitely supported discrete part: Propagator and explicit solution |
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501 | (2) |
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Physical interpretations in the quantum-mechanical case |
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503 | (1) |
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Physical interpretations in the diffusion case |
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504 | (1) |
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Further connections with Feynman's Operational calculus |
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505 | (2) |
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The Feynman--Kac formula with a Lebesgue--Stieltjes measure: The general case (arbitrary measure η) |
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507 | (23) |
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Integral equation (integrated form of the evolution equation) |
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507 | (2) |
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Basic properties of the solution to the integral equation |
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509 | (2) |
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Quantum-mechanical case: Reformulation in the interaction (or Dirac) picture |
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511 | (3) |
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Product integral representation of the solution |
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514 | (3) |
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Distributional differential equation (true differential form of the evolution equation) |
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517 | (2) |
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519 | (1) |
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Scattering matrix and improper product integral |
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520 | (1) |
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Sketch of the proof of the integral equation |
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521 | (9) |
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Noncommutative operations on Wiener functionals, disentangling algebras and Feynman's operational calculus |
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530 | (32) |
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530 | (2) |
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Preliminaries: maps, measures and measurability |
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532 | (3) |
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The noncommutative operations * and + |
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535 | (5) |
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The functional integrals Ktλ(·) and the operations * and + |
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540 | (4) |
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The disentangling algebras At, the operations * and +, and the disentangling process |
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544 | (9) |
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Examples: Trigonometric, binomial and exponential formulas |
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552 | (1) |
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Appendix: Quantization, axiomatic Feynman's operational calculus, and generalized functional integral |
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553 | (9) |
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Algebraic and analytic axioms |
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554 | (2) |
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Consequences of the axioms |
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556 | (3) |
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Examples: the disentangling algebras and analytic Feynman integrals |
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559 | (3) |
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Feynman's operational calculus and evolution equations |
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562 | (47) |
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Introduction and hypotheses |
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562 | (6) |
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Feynman's operational calculus as a generalized path integral |
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562 | (1) |
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Exponentials of sums of noncommuting operators |
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563 | (1) |
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Disentangling exponentials of sums via perturbation series |
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563 | (2) |
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Local and nonlocal potentials |
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565 | (1) |
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566 | (2) |
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Disentangling exp{--tα + ∫t0 β(s)μ(ds)} |
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568 | (5) |
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Disentangling exp{-tα + ∫t0 β1(s)μ1(ds) + ... + ∫t0 βn(s)μn(ds)} |
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573 | (8) |
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Convergence of the disentangled series |
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581 | (6) |
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587 | (9) |
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Uniqueness of the solution to the evolution equation |
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596 | (3) |
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Further examples of the disentangling process |
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599 | (10) |
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Nonlocal potentials relevant to phenomenological nuclear theory |
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604 | (5) |
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Further work on or related to the Feynman integral |
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609 | (88) |
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Transform approaches to the Feynman integral. References to further approaches |
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609 | (28) |
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The Fresnel integral and other transform approaches to the Feynman integral |
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610 | (1) |
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610 | (1) |
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Properties of the Fresnel integral |
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611 | (2) |
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An approach to the Feynman integral via the Fresnel integral |
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613 | (1) |
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Advantages and disadvantages of Fresnel integral approaches to the Feynman integral |
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613 | (2) |
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615 | (1) |
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The Poison process and transforms |
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616 | (1) |
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A ``Fresnel integral'' on classical Wiener space |
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616 | (4) |
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The Banach algebras S and F(H1) are the same |
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620 | (1) |
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Consequences of the close relationship between S and F(H1) |
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621 | (1) |
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622 | (1) |
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A unified theory of Fresnel integrals: Introductory remarks |
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623 | (1) |
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624 | (2) |
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A unified theory of Fresnel integrals (continued) |
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626 | (3) |
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The Fresnel classes along with quadratic forms |
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629 | (1) |
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The classes Gq(H) and Gq(B) |
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630 | (1) |
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631 | (1) |
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Functions in the Fresnel class of an abstract Wiener space: Examples of abstract Wiener spaces |
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632 | (4) |
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Fourier--Feynman transforms, convolution, and the first variation for functions in S |
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636 | (1) |
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References to further approaches to the Feynman integral |
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636 | (1) |
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The influence of heuristic Feynman integrals on contemporary mathematics and physics: Some examples |
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637 | (60) |
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The heuristic Feynman path integral |
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638 | (1) |
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Knot invariants and low-dimentional topology |
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639 | (1) |
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The Jones polynomial invariant for knots and links |
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639 | (2) |
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Witten's topological invariants via Feynman path integrals |
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641 | (13) |
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Further developments: Vassiliev invariants and the Kontsevich integral |
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654 | (5) |
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Further comments and references on subjects related to the Feynman integral |
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659 | (1) |
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Supersymmetric Feynman path integrals and the Atiyah--Singer index theorem |
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659 | (15) |
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Deformation quantization: Star products and perturbation series |
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674 | (8) |
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Gauge field theory and Feynman path integrals |
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682 | (6) |
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String theory, Feynman--Polyakov integrals, and dualities |
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688 | (7) |
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What lies ahead? Towards a geometrization of Feynman path integrals? |
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695 | (2) |
| References |
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697 | (48) |
| Index of symbols |
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745 | (5) |
| Author index |
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750 | (6) |
| Subject index |
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756 | |
