viii
Contents
3 Berggren Basis and Completeness Relations. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 81
3.1 Normalization of Gamow Functions .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 82
3.2 One-Body Completeness Relation .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 88
3.2.1 One-Body Completeness Relation for = 0 Neutrons . . . . 89
3.2.2 Completeness Relation of Coulomb Wave Functions . . . . . . 92
3.2.3 Completeness Relation for the General Case . . . . . . . . . . . . . . . 94
3.3 Normalization and Orthogonality of Gamow States
and One-Body Matrix Elements . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 98
3.4 Matrix Elements Involving Scattering States . . . . .. . . . . . . . . . . . . . . . . . . . 102
3.5 Completeness Relation Involving Single-Particle Gamow States . . . 103
3.5.1 Domain of Applicability of the Berggren
Completeness Relation . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 104
3.5.2 Berggren Completeness Relation for Complex
Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 106
3.5.3 Overcompleteness Relations Involving
Single-Particle Gamow States . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 109
3.6 Newton and Berggren Completeness Relations Generated
by Nonlocal Potentials.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 110
3.7 Numerical Implementation of the Berggren Completeness
Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 112
3.7.1 Completeness Relations Involving Proton States . . . . . . . . . . . 113
3.7.2 Completeness Relations Involving Antibound s 1/2
States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 123
3.8 Complex-Symmetric Operators and Matrices . . . .. . . . . . . . . . . . . . . . . . . . 127
3.8.1 Diagonalizability of Complex-Symmetric Matrices. . . . . . . . 129
3.8.2 The Two-Dimensional Complex Symmetric Matrix . . . . . . . 129
3.8.3 Numerical Studies of Complex-Symmetric Matrices . . . . . . 131
Solutions to Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 134
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 144
4 Two-Particle Systems in the Berggren Basis . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 147
4.1 Exact Formulation of the Two-Particle Problem Using
Relative Coordinates.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 147
4.2 Study of Two-Nucleon Systems with Realistic Interactions .. . . . . . . . 149
4.2.1 Numerical Studies of Dineutron, Diproton, and
Deuteron in Berggren Basis . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 152
4.3 The Particle-Rotor Model in the Berggren Basis. . . . . . . . . . . . . . . . . . . . . 157
4.3.1 Mathematical Formalism of the Particle-Rotor Model .. . . . 159
4.3.2 Dipolar Anions .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 162
4.3.3 Quadrupolar Anions . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 167
4.3.4 Weakly Bound and Unbound Atomic Nuclei .. . . . . . . . . . . . . . 173
Solutions to Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 180
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 182
Contents
3 Berggren Basis and Completeness Relations. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 81
3.1 Normalization of Gamow Functions .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 82
3.2 One-Body Completeness Relation .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 88
3.2.1 One-Body Completeness Relation for = 0 Neutrons . . . . 89
3.2.2 Completeness Relation of Coulomb Wave Functions . . . . . . 92
3.2.3 Completeness Relation for the General Case . . . . . . . . . . . . . . . 94
3.3 Normalization and Orthogonality of Gamow States
and One-Body Matrix Elements . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 98
3.4 Matrix Elements Involving Scattering States . . . . .. . . . . . . . . . . . . . . . . . . . 102
3.5 Completeness Relation Involving Single-Particle Gamow States . . . 103
3.5.1 Domain of Applicability of the Berggren
Completeness Relation . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 104
3.5.2 Berggren Completeness Relation for Complex
Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 106
3.5.3 Overcompleteness Relations Involving
Single-Particle Gamow States . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 109
3.6 Newton and Berggren Completeness Relations Generated
by Nonlocal Potentials.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 110
3.7 Numerical Implementation of the Berggren Completeness
Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 112
3.7.1 Completeness Relations Involving Proton States . . . . . . . . . . . 113
3.7.2 Completeness Relations Involving Antibound s 1/2
States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 123
3.8 Complex-Symmetric Operators and Matrices . . . .. . . . . . . . . . . . . . . . . . . . 127
3.8.1 Diagonalizability of Complex-Symmetric Matrices. . . . . . . . 129
3.8.2 The Two-Dimensional Complex Symmetric Matrix . . . . . . . 129
3.8.3 Numerical Studies of Complex-Symmetric Matrices . . . . . . 131
Solutions to Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 134
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 144
4 Two-Particle Systems in the Berggren Basis . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 147
4.1 Exact Formulation of the Two-Particle Problem Using
Relative Coordinates.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 147
4.2 Study of Two-Nucleon Systems with Realistic Interactions .. . . . . . . . 149
4.2.1 Numerical Studies of Dineutron, Diproton, and
Deuteron in Berggren Basis . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 152
4.3 The Particle-Rotor Model in the Berggren Basis. . . . . . . . . . . . . . . . . . . . . 157
4.3.1 Mathematical Formalism of the Particle-Rotor Model .. . . . 159
4.3.2 Dipolar Anions .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 162
4.3.3 Quadrupolar Anions . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 167
4.3.4 Weakly Bound and Unbound Atomic Nuclei .. . . . . . . . . . . . . . 173
Solutions to Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 180
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 182