Diagonalizing Quadratic Bosonic Operators by Non-Autonomous Flow Equations.
- 1st ed.
- 1 online resource (134 pages)
- Memoirs of the American Mathematical Society Series ; v.240 .
- Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Chapter I. Introduction -- Chapter II. Diagonalization of Quadratic Boson Hamiltonians -- II.1. Quadratic Boson Operators -- II.2. Main Results -- II.3. Historical Overview -- Chapter III. Brocket-Wegner Flow for Quadratic Boson Operators -- III.1. Setup of the Brocket-Wegner Flow -- III.2. Mathematical Foundations of our Method -- III.3. Asymptotic Properties of the Brocket-Wegner Flow -- Chapter IV. Illustration of the Method -- IV.1. The Brocket-Wegner Flow on Bogoliubov's Example -- IV.2. Blow-up of the Brocket-Wegner Flow -- Chapter V. Technical Proofs on the One-Particle Hilbert Space -- V.1. Well-Posedness of the Flow -- V.2. Constants of Motion -- V.3. Asymptotics Properties of the Flow -- Chapter VI. Technical Proofs on the Boson Fock Space -- VI.1. Existence and Uniqueness of the Unitary Propagator -- VI.2. Brocket-Wegner Flow on Quadratic Boson Operators -- VI.3. Quasi -Diagonalization of Quadratic Boson Operators -- VI.4. -Diagonalization of Quadratic Boson Operators -- Chapter VII. Appendix -- VII.1. Non-Autonomous Evolution Equations on Banach Spaces -- VII.2. Autonomous Generators of Bogoliubov Transformations -- VII.3. Trace and Representation of Hilbert-Schmidt Operators -- References -- Back Cover.
The authors study a non-autonomous, non-linear evolution equation on the space of operators on a complex Hilbert space. They specify assumptions that ensure the global existence of its solutions and allow them to derive its asymptotics at temporal infinity. They demonstrate that these assumptions are optimal in a suitable sense and more general than those used before. The evolution equation derives from the Brocketâe"Wegner flow that was proposed to diagonalize matrices and operators by a strongly continuous unitary flow. In fact, the solution of the non-linear flow equation leads to a diagonalization of Hamiltonian operators in boson quantum field theory which are quadratic in the field.