Below you find a list of my publications with links to the arxiv versions. Please consult my research overview (as PDF) for short explanations of my results.

## 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. Our results constitutes a first step towards a rigorous justification of the random-phase approximation (RPA). Our approach is based on approximate bosonization of collective particle-hole excitations, similar as in the plasmon theory of Bohm and Pines.

- 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]; accepted for publication in
*Commun. Math. Phys.*(2019)

## 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. We obtain a partial verification of Dyson's claim that spin-wave interactions have extremely small effects at low temperature.

- 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. The integral scheme is implemented in the CP2K quantum chemistry software, leading to a three orders of magnitude speed-up compared to previous methods.

- 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; also called the time-dependent BCS equations, describing Cooper pairs in superconductivity) and the bosonic Hartree-Fock-Bogoliubov equations from a reformulation of the time-dependent variational principle, proving optimality of these approximations. We also give a proof of well-posedness for the Bogoliubov-de-Gennes equations with singular interactions.

- 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 time-dependent Hartree-Fock equation (TDHF) 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, e.g., initial data prepared at positive temperature. As a second step of approximation, we derive the Vlasov equation of kinetic theory.

- 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 non-equilibrium properties 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 non-relativistic quantum electrodynamics:

- N. Benedikter: Dynamics of the Radiative Decay of Excited Atoms (2010), Diploma thesis at the University of Stuttgart