Mathematical and Numerical Aspects of Quantum Dynamics

On effective equations of quantum physics

Israel Michael Sigal

University of Toronto


The Hartree-Fock (HF) equation is a (if not the) key effective equation of quantum physics. It gives a fairly accurate and yet sufficiently simple description of large (and not so large) systems of quantum particles. The trade-off here is the nonlinearity for high dimensions: while the n-particle Schrödinger equation is a linear equation in 3n space dimensions, the Hartree-Fock one is a nonlinear one in 3 space dimensions. It was discovered by Bardeen, Cooper and Schrieffer for fermions and by Bogolubov, for bosons, that for quantum fluids (superconductors and superfluids, respectively):

  • the HFE falls short
  • there is a generalization of the HFE describing these phenomena.
It turns out that this generalization is mathematically very natural and leads to the (time-dependent) Bogolubov-de Gennes (BdG) or Hartree-Fock-Bogolubov (HFB) equations, depending on whether we deal with fermions or bosons. (The BdG equations give an equivalent formulation of the BCS theory of superconductivity for which Bardeen, Cooper and Schrieffer received the Nobel prize.) There are many fundamental problems about these equations which are completely open. In this talk, I review some recent results about these equations. I will explain how these equations emerge from the N-body Schr\”odinger equations and will discuss their general features and key physical classes of stationary solutions. I will also touch upon the density functional theory which is a natural modification of the HF equations.

The talk is based on joint work with Ilias Chenn and with V. Bach, S. Breteaux, Th. Chen and J. Fröhlich.