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Diagonalizing quadratic bosonic operators by non-autonomous flow equation

About this Title

Volker Bach and Jean-Bernard Bru

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 240, Number 1138
ISBNs: 978-1-4704-1705-5 (print); 978-1-4704-2828-0 (online)
Published electronically: November 13, 2015
Keywords:Quadratic operators, flow equations for operators, evolution equations, Brocket–Wegner flow, double bracket flow

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Diagonalization of Quadratic Boson Hamiltonians
  • Chapter 3. Brocket–Wegner Flow for Quadratic Boson Operators
  • Chapter 4. Illustration of the Method
  • Chapter 5. Technical Proofs on the One–Particle Hilbert Space
  • Chapter 6. Technical Proofs on the Boson Fock Space
  • Chapter 7. Appendix


We study a non–autonomous, non-linear evolution equation on the space of operators on a complex Hilbert space. We specify assumptions that ensure the global existence of its solutions and allow us to derive its asymptotics at temporal infinity. We demonstrate that these assumptions are optimal in a suitable sense and more general than those used before. The evolution equation derives from the Brocket–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.

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