The derivation of effective evolution equations is central to the study of non-stationary quantum many-body systems, and widely used in contexts such as superconductivity, nuclear physics, Bose-Einstein condensation and quantum chemistry. We reformulate the Dirac-Frenkel approximation principle in terms of reduced density matrices and apply it to fermionic and bosonic many-body systems. We obtain the Bogoliubov-de Gennes and Hartree-Fock-Bogoliubov equations, respectively. While we do not prove quantitative error estimates, our formulation does show that the approximation is optimal within the class of quasifree states. Furthermore, we prove well-posedness of the Bogoliubov-de Gennes equations in energy space and discuss conserved quantities.

The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de Gennes Equations / N. Benedikter, J. Sok, J. Solovej. - In: ANNALES HENRI POINCARE'. - ISSN 1424-0637. - 19:4(2018), pp. 1167-1214. [10.1007/s00023-018-0644-z]

The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de Gennes Equations

N. Benedikter;
2018

Abstract

The derivation of effective evolution equations is central to the study of non-stationary quantum many-body systems, and widely used in contexts such as superconductivity, nuclear physics, Bose-Einstein condensation and quantum chemistry. We reformulate the Dirac-Frenkel approximation principle in terms of reduced density matrices and apply it to fermionic and bosonic many-body systems. We obtain the Bogoliubov-de Gennes and Hartree-Fock-Bogoliubov equations, respectively. While we do not prove quantitative error estimates, our formulation does show that the approximation is optimal within the class of quasifree states. Furthermore, we prove well-posedness of the Bogoliubov-de Gennes equations in energy space and discuss conserved quantities.
gross-Potaevskii equation; nonlinear Schrodinger-equation;mean-field limit; rigorous derivation; interacting bosons; pair excitations; dynamics; existence; approzimation; evolution
Settore MAT/07 - Fisica Matematica
2018
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/703462
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