Spin Wave Theory
We study corrections to the free energy in the Quantum Heisenberg Ferromagnet due to remainder interaction effects in the Spin Wave Theory.
- N. Benedikter: Interaction Corrections to Spin-Wave Theory in the Quantum Heisenberg Ferromagnet, Mathematical Physics, Analysis, and Geometry 20, 1-21 (2017)
Efficient Evaluation of Solid Harmonic Gaussian Integrals
We derive explicit formulas for certain integrals in numerical quantum chemistry calculations.
- D. Golze, N. Benedikter, M. Iannuzzi, J. Wilhelm, and J. Hutter: Fast evaluation of solid harmonic Gaussian integrals for local resolution-of-the-identity methods and range-separated hybrid functionals, Journal of Chemical Physics 146, 034105 (2017)
Effective Evolution Equations
We derive the fermionic Bogoliubov-de-Gennes equations (Hartree-Fock equations with pairing density) and the bosonic Hartree-Fock-Bogoliubov equations from a reformulation of the geometric Dirac-Frenkel principle, proving optimality of these approximations. We also give a proof of well-posedness for the Bogoliubov-de-Gennes equations.
- N. Benedikter, J. Sok, and J.P. Solovej: The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de-Gennes Equations, Preprint arXiv:1706.03082 [math-ph]
In the following lecture notes we discuss a wide range of results concerning effective evolution equations for bosonic and fermionic systems.
- N. Benedikter, M. Porta, and B. Schlein: Effective Evolution Equations from Quantum Dynamics (2016), in SpringerBriefs in Mathematical Physics
The many-body Schrödinger equation in certain scaling regimes gives rise to effective nonlinear dynamics. An overview can be found in my thesis:
- N. Benedikter: Effective Evolution Equations from Many-Body Quantum Mechanics, (2014) Thesis University of Bonn
Fermionic Effective Evolution Equations
We derive the Hartree-Fock equation as governing the effective dynamics of fermions in the mean-field regime. In a recent paper, we extend the derivation to mixed states as initial data. As a second step of approximation, we derive the Vlasov equation.
- N. Benedikter, M. Porta. C. Saffirio, and B. Schlein: From the Hartree dynamics to the Vlasov equation, ARMA 221, 273-334 (2016)
- N. Benedikter, V. Jaksic, M. Porta, C. Saffirio, and B. Schlein: Mean-field Evolution of Fermionic Mixed States, Comm. Pure Appl. Math. 69, 2250-2303 (2014)
- N. Benedikter, M. Porta, and B. Schlein: Hartree-Fock dynamics for weakly interacting fermions (2014), in Proceedings of the QMath12 Conference
- N. Benedikter, M. Porta, and B. Schlein: Mean-Field Dynamics of Fermions with Relativistic Dispersion, J. Math. Phys. 55, 021901 (2014)
- N. Benedikter, M. Porta, and B. Schlein: Mean-field Evolution of Fermionic Systems, Comm. Math. Phys. 331, 1087-1131 (2014)
Bosonic Effective Evolution Equation
We derive the Gross-Pitaevskii equation describing the dynamics of dilute Bose-Einstein condensates:
- N. Benedikter: Deriving the Gross-Pitaevskii Equation (2014), in Proceedings of the QMath12 Conference
- N. Benedikter, G. de Oliveira, and B. Schlein: Quantitative Derivation of the Gross-Pitaevskii Equation, Comm. Pure Appl. Math. 68, 1399-1482 (2014)
Physical experience shows that excited atoms relax to the ground state by emission of photons. We study the rate of relaxation in models of non-relativistic quantum electrodynamics:
- N. Benedikter: Dynamics of the Radiative Decay of Excited Atoms (2010), Diploma thesis at the University of Stuttgart