This book presents new developments in the open quantum systems theory with emphasis on applications to the (frequent) measurement theory. In the first part of the book, the uniqueness theorems for the solutions to the restricted Weyl commutation relations braiding unitary groups and semi-groups of contractions are discussed. The major theme involves an intrinsic characterization of the simplest symmetric operator solutions to the Heisenberg uncertainty relations, the problem posed by Jørgensen and Muhly, followed by the proof of the uniqueness theorems for the simplest solutions to the restricted Weyl commutation relations. The detailed study of unitary invariants of the corresponding dissipative and symmetric operators opens up a look at the classical Stone-von Neumann uniqueness theorem from a new angle and provides an extended version of the uniqueness result relating various realizations of a differentiation operator on the corresponding metric graphs. The second part of the book is devoted to mathematical problems of the quantum measurements under continuous monitoring. Among the topics discussed are the complementarity of the Quantum Zeno effect and Exponential Decay scenario in frequent quantum measurements, and a rigorous treatment, within continuous monitoring paradigm, of the celebrated "double-slit experiment" where the renowned exclusive and interference measurement alternatives approach in quantum theory is presented in a way that is accessible for mathematicians. One of the striking applications of the generalized (1-stable) central limit theorem is the mathematical evidence of exponential decay of unstable states of the quantum pendulum under continuous monitoring.
Author(s): Konstantin A. Makarov, Eduard Tsekanovskii
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 299
City: Singapore
CONTENTS
Acknowledgments
Preface
Part 1: Representations of Operator Commutation Relations
1. Introduction
2. Preliminaries and Basic Definitions
3. The Commutation Relations and Character-Automorphic Functions
4. The Differentiation Operator on Metric Graphs
5. The Magnetic Hamiltonian
6. The Livšic Function s(D,DΘ)(z)
7. The Weyl-Titchmarsh Function M(D,DΘ)(z)
8. The Model Dissipative Operators
9. The Characteristic Function S(D,D,DΘ)(z)
10. The Transmission Coefficient and the Characteristic Function
11. Uniqueness Results
12. Dissipative Solutions to the CCR
13. Main Results
14. Unitary Dynamics on the Full Graph
Part 2: Continuous Monitoring, Quantum Measurements
15. Continuous Monitoring of the Quantum Systems
15.1. Quantum Zeno Effect
15.2. Anti-Zeno Effect
15.3. The Exponential Decay
15.4. Frequent Measurements and the Time-Energy Uncertainty Principle
15.5. Presto
16. The Quantum Zeno versus Anti-zeno Effect Alternative
17. The Quantum Zeno Effect versus Exponential Decay Alternative
18. Preliminaries: Probabilities versus Amplitudes
19. Massless Particles on a Ring
20. Continuous Monitoring with Interference
20.1. Discussion
21. Continuous Monitoring with No Interference
21.1. Discussion
22. The Self-adjoint Dilation
23. General Open Quantum Systems on a Ring
23.1. Discussion
23.2. Random Phase Method
24. Operator Coupling Limit Theorems
Appendix A The Characteristic Function for Rank-One Perturbations
Appendix B Prime Symmetric Operators
Appendix C A Functional Model of a Triple
Appendix D The Spectral Analysis of the Model Dissipative Operator
Appendix E Transformation Laws
Appendix F The Invariance Principle
Appendix G The Operator Coupling and the Multiplication Theorem
Appendix H Stable Laws
References
Index
