## Correlation Energy of the Fermi Gas

We derive a formula of the type proposed by Gell-Mann and Brueckner as an upper bound for the correlation energy of the Fermi gas in the mean-field scaling regime.

- N. Benedikter, P. T. Nam, M. Porta, B. Schlein, and R. Seiringer: Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime, arXiv:1809.01902 [math-ph]

## 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 in the Large-S Limit of 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,
*Annales Henri Poincaré*,**19**(4), 1167-1214 (2018)

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)

## Quantum Electrodynamics

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