• Corpus ID: 119175840

On the Bogolubov-de Gennes Equations

@article{Chenn2017OnTB,
  title={On the Bogolubov-de Gennes Equations},
  author={Ilias Chenn and Israel Michael Sigal},
  journal={arXiv: Mathematical Physics},
  year={2017}
}
We consider the Bogolubov-de Gennes equations giving an equivalent formulation of the BCS theory of superconductivity. We are interested in the case when the magnetic field is present. We (a) discuss their general features, (b) isolate key physical classes of solutions (normal, vortex and vortex lattice states) and (c) prove existence of the normal, vortex and vortex lattice states and stability/instability of the normal states for large/small temperature or/and magnetic fields. 
View PDF on arXiv
6 Citations
Background Citations
2

6 Citations

On the Hartree-Fock-Bogoliubov equations

We review some results of our paper arXiv:1602.05171v2 on the "nonlinear quasifree approximation" to the many-body Schr\"odinger dynamics of Bose gases. In that paper, we derive, with the help of…

The Dirac–Frenkel Principle for Reduced Density Matrices, and the Bogoliubov–de Gennes Equations

The derivation of effective evolution equations is central to the study of non-stationary quantum many-body systems, and widely used in contexts such as superconductivity, nuclear physics,…

A lower bound for the BCS functional with boundary conditions at infinity

We consider a many-body system of fermionic atoms interacting via a local pair potential and subject to an external potential within the framework of Bardeen-Cooper-Schrieffer (BCS) theory. We…

On Derivation of the Poisson–Boltzmann Equation

Starting from the microscopic reduced Hartree–Fock equation, we derive the macroscopic linearized Poisson–Boltzmann equation for the electrostatic potential associated with the electron density.

Differential equations of quantum mechanics

  • I. Sigal
  • Physics, Mathematics
    Quarterly of Applied Mathematics
  • 2022
We review very briefly the main mathematical structures and results in some important areas of Quantum Mechanics involving PDEs and formulate open problems.

On Derivation of the Poisson–Boltzmann Equation

Starting from the microscopic reduced Hartree–Fock equation, we derive the macroscopic linearized Poisson–Boltzmann equation for the electrostatic potential associated with the electron density.

References

SHOWING 1-10 OF 14 REFERENCES

Magnetic Vortices, Abrikosov Lattices, and Automorphic Functions

We address the macroscopic theory of superconductivity - the Ginzburg-Landau theory. This theory %Macroscopic theory of superconductivity is based on the celebrated Ginzburg - Landau equations. First…

The Bardeen–Cooper–Schrieffer functional of superconductivity and its mathematical properties

We review recent results concerning the mathematical properties of the Bardeen–Cooper–Schrieffer (BCS) functional of superconductivity, which were obtained in a series of papers, partly in…

Microscopic Derivation of the Ginzburg-Landau Model

We present a summary of our recent rigorous derivation of the celebrated Ginzburg–Landau (GL) theory, starting from the microscopic Bardeen–Cooper–Schrieffer (BCS) model. Close to the critical…

Ginzburg-Landau theory for superconductors

In this review the author describes how the simple Ginzburg-Landau approach lies in the heart of the general theory of superconductors. The reader is introduced to the handling of the theory and to…

The BCS Functional for General Pair Interactions

The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous…

Generalized Hartree-Fock theory and the Hubbard model

The familiar unrestricted Hartree-Fock variational principles is generalized to include quasi-free states. As we show, these are in one-to-one correspondence with the one-particle density matrices…

Translation-invariant quasi-free states for fermionic systems and the BCS approximation

We study translation-invariant quasi-free states for a system of fermions with two-particle interactions. The associated energy functional is similar to the BCS functional but also includes direct…

Superconductivity of metals and alloys

Magnetic vortices

In spaces with constant curvature, magnetic trajectories of a Killing magnetic field are showed to be centerlines of Kirchhoff elastic rods. If the space is also simply connected, then a Killing…

Separation of center of mass in homogeneous magnetic fields