Niels Benedikter

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.

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.

Efficient Evaluation of Solid Harmonic Gaussian Integrals

We derive explicit formulas for certain integrals in numerical quantum chemistry calculations.

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.

In the following lecture notes we discuss a wide range of results concerning effective evolution equations for bosonic and fermionic systems.

The many-body Schrödinger equation in certain scaling regimes gives rise to effective nonlinear dynamics. An overview can be found in my thesis:

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.

Bosonic Effective Evolution Equation

We derive the Gross-Pitaevskii equation describing the dynamics of dilute Bose-Einstein condensates:

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: