Lagrangian Analysis and Quantum Mechanics: A Mathematical Structure Related to Asymptotic Expansions and the Maslov Index

Lagrangian Analysis and Quantum Mechanics......Page 1
ISBN 0262120879......Page 2
Dedication......Page 3
Contents......Page 5
Preface......Page 9
Index of Symbols......Page 11
Index of Concepts......Page 15
1. The Metaplectic Group Mp(l)......Page 17
2. The Subgroup Sp_2(l) of MP(l)......Page 25
3. Differential Operators with Polynomial Coefficients......Page 36
0. Introduction......Page 41
1. Choice of Hermitian Structures on Z(1)......Page 42
2. The Lagrangian Grassmannian A(l) of Z(1)......Page 43
3. The Covering Groups of Sp(l) and the Covering Spaces of A(l)......Page 47
4. Indices of Inertia......Page 53
5. The Maslov Index m on A2 (1)......Page 58
6. The Jump of the Maslov Index m(A., A..) at a Point (ti, A.\') Where dim;, n A.\' = 1......Page 63
7. The Maslov Index on Spa (l); the Mixed Inertia......Page 67
8. Maslov Indices on A,(/) and Sp,,(/)......Page 69
9. Lagrangian Manifolds......Page 71
10. q-Orientation (q = 1, 2, 3, ..., cc)......Page 72
1. Symplectic Space Z......Page 74
2. The Frames of Z......Page 76
3. The q-Frames of Z......Page 77
Conclusion......Page 81
Introduction......Page 83
1. The Algebra W(X) of Asymptotic Equivalence Classes......Page 84
2. Formal Numbers; Formal Functions......Page 89
3. Integration of Elements of .FO(X)......Page 96
4. Transformation of Formal Functions by Elements of Sp2(l)......Page 102
5. Norm and Scalar product of Formal Functions with Compact Support......Page 107
6. Formal Differential Operators......Page 113
7. Formal Distributions......Page 118
0. Summary......Page 120
1. Lagrangian Operators......Page 121
2. Lagrangian Functions on V......Page 125
3. Lagrangian Functions on V......Page 131
5. Lagrangian Distributions......Page 139
1. Lagrangian Manifolds on Which Lagrangian Solutions of aU = 0 Are Defined......Page 140
2. Review of E. Cartan\'s Theory of Pfaffian Forms......Page 141
3. Lagrangian Manifolds in the Symplectic Space Z and in Its Hypersurfaces......Page 145
4. Calculation of aU......Page 151
5. Resolution of the Lagrangian Equation aU = 0......Page 155
6. Solutions of the Lagrangian Equation aU = 0 mod(1/v2) with Positive Lagrangian Amplitude: Maslov\'s Quantization......Page 159
7. Solution of Some Lagrangian Systems in One Unknown......Page 161
Conclusion......Page 167
1. Calculation of Em_a\' U,,......Page 168
2. Resolution of the Lagrangian System aU = 0 in Which the Zeros of det ao Are Simple Zeros......Page 172
3. A Special Lagrangian System aU = 0 in Which the Zeros of det ao Are Multiple Zeros......Page 175
Introduction......Page 179
1. Four Functions Whose Pairs Are All in Involution on E3 Q+ E3 Except for One......Page 182
2. Choice of a Hamiltonian H......Page 186
3. The Quantized Tori T(l, m, n) Characterizing Solutions, Defined mod(1/v) on Compact Manifolds, of the Lagrangian System aU = (aL2 - L2)U = (am - Mo)U = 0 mod (1/v2)......Page 190
4. Examples: The Schrodinger and Klein-Gordon Operators......Page 195
0. Introduction......Page 200
1. Solutions of the Equation aU = 0 mod(1/v2) with Lagrangian Amplitude >,0 Defined on the Tori V[L0, M0]......Page 201
2. Compact Lagrangian Manifolds V, Other Than the Tori V[L0, M0], on Which Solutions of the Equation aU = 0 mod(1/v2) with Lagrangian Amplitude 30 Exist......Page 206
3. Example: The Schrodinger-Klein-Gordon Operator......Page 220
0. Introduction......Page 223
2. Case of an Operator a Commuting with aL2 and am 207......Page 226
3. A Special Case......Page 237
4. The Schrodinger-Klein-Gordon Case......Page 242
0. Introduction......Page 246
1. Study of Problem (0.1) without Assumption (0.4)......Page 247
2. The Schrodinger-Klein-Gordon Case......Page 250
Conclusion......Page 253
1. Choice of Operators Commuting mod(1!v3)......Page 254
2. Resolution of a Lagrangian Problem in Two Unknowns......Page 256
1. Reduction of the Dirac Equation in Lagrangian Analysis......Page 264
2. The Reduced Dirac Equation for a One-Electron Atom in a Constant Magnetic Field......Page 270
3. The Energy Levels......Page 274
4. Crude Interpretation of the Spin in Lagrangian Analysis......Page 278
5. The Probability of the Presence of the Electron......Page 280
Conclusion......Page 282
Bibliography......Page 285