Calculus and Mechanics on Two-Point Homogenous Riemannian Spaces (eBook)

Alexey V. Shchepetilov (Autor)

eBook Download: PDF

2006 | 1. Auflage
274 Seiten
Springer-Verlag
978-3-540-35386-7 (ISBN)

Infos

The present monograph gives a short and concise introduction to classical and quantum mechanics on two-point homogenous Riemannian spaces, with empahsis on spaces with constant curvature. Chapter 1-4 provide the basic notations from differential geometry for studying two-body dynamics in these spaces. Chapter 5 deals with the problem of finding explicitly invariant expressions for the two-body quantum Hamiltonian. Chapter 6 addresses one-body problems in a central potential. Chapter 7 studies the classical counterpart of the quantum system of chapter 5. Chapter 8 investigates some applications in the quantum realm, namely for the coulomb and oscillator potentials.



Keywords: Hamiltonian functions, Riemannian spaces, integrable systems, two-body problem.

Inhalt

Preface 6
Contents 12
Glossary 16
1 Two-Point Homogeneous Riemannian Spaces 19
1.1 Classification 19
1.2 Special Expansion of the Lie Algebra of Infinitesimal Isometries for Two- Point Homogeneous Riemannian Spaces 21
1.3 Models of Classical Compact Two-Point Homogeneous Riemannian Spaces 26
1.4 The Model of the Projective Cayley Plane 33
2 Differential Operators on Smooth Manifolds 41
2.1 Invariant Differential Operators on Lie Groups and Homogeneous Manifolds 41
2.2 Laplace–Beltrami Operator in a Moving Frame 53
2.3 Self-Adjointness of Hamiltonians 55
2.4 General Scheme of Quantum-Mechanical Reduction 65
3 Algebras of Invariant Differential Operators on Unit Sphere Bundles Over Two- Point Homogeneous Riemannian Spaces 69
3.1 Invariant Differential Operators on 69
3.2 Algebras Di.I(Pn(H)S) and Di.I(Hn(H)S) 72
3.3 Algebras Di.I(Pn(C)S) and Di.I(Hn(C)S) 84
3.4 Algebras Di.I(Pn(R)S), Di.I(Sn S) and Di.I(Hn(R)S) 89
3.5 Algebras Di.I(Pn(Ca)S) and Di.I(Hn(Ca)S) 93
3.6 The Kernel of the Operator 102
4 Hamiltonian Systems with Symmetry and Their Reduction 105
4.1 Basic Facts from Hamiltonian Mechanics 105
4.2 Hamiltonian Mechanics with Symmetry 109
4.3 Hamiltonian Systems on Cotangent Bundles 116
5 Two-Body Hamiltonian on Two-Point Homogeneous Spaces 131
5.1 Homogeneous Submanifolds in the Con.guration Space of the Two- Body Problem 131
5.2 Two-Body Hamiltonian on a Compact Two-Point Homogeneous Space 134
5.3 Two-Body Hamiltonian on a Noncompact Two-Point Homogeneous Space 140
5.4 Connection of the Two-Body Hamiltonian and the Algebra Diff G(QS) 141
6 Particle in a Central Field on Two-Point Homogeneous Spaces 145
6.1 Integrability of the One-Particle Motion in a Central Field on Two- Point Homogeneous Spaces 145
6.2 Particle Motion in Bertrand Potentials on Constant Curvature Spaces 148
6.3 Quantum Mechanical One-Body Problem for Bertrand Potentials on Constant Curvature Spaces 160
6.4 The History of the Problem of One and Two Particles in a Central Field on Constant Curvature Spaces 173
7 Classical Two-Body Problem on Two-Point Homogeneous Riemannian Spaces 179
7.1 Explicitly Invariant Form of the Hamiltonian Two- Body Function for Compact Two- Point Homogeneous Spaces 179
7.2 Explicitly Invariant Form of the Hamiltonian Two- Body Function for Noncompact Two- Point Homogeneous Spaces 184
7.3 Dynamics of the Two-Body System and the Problem of Particles’ Collision 189
7.4 The Center of Mass Problem on Two-Point Homogeneous Spaces 194
7.5 Hamiltonian Reduction of the Two-Body Problem on Constant Curvature Spaces 200
8 Quasi-Exactly Solvability of the Quantum Mechanical Two- Body Problem on Spheres 209
8.1 Regular Representations of Compact Lie Groups 210
8.2 Common Eigenfunctions of Operators for Spheres Sn and Projective Spaces Pn(R) 212
8.3 Scalar Spectral Equations and Some Energy Levels for the Two- Body Problem 224
8.4 The Problem of the Discrete Spectrum on Noncompact Spaces 234
A Calculations of Commutator Relations for Algebras of Invariant Differential Operator 237
B Some Fuchsian Differential Equations 243
C Orthogonal Complex Lie Algebras and Their Representations 251
C.1 Lie Algebra 251
C.2 Lie Algebra 253
C.3 Restrictions of 254
and 254
Representations 254
C.4 The Proof of Two Expansions 255
D Unsolved Problems 259
References 261
Index 271

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Produkt-Details

• Erscheint lt. Verlag: 1.1.2006
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• ISBN-10: 3-540-35386-0 / 3540353860
• ISBN-13: 978-3-540-35386-7 / 9783540353867