Professor of Mathematics

Michael Melgaard Department of Mathematics
School of Mathematical and Physical Sciences
University of Sussex
Falmer Campus
Brighton BN1 9QH
Great Britain

Funding

Two Ph.D. opportunities within Analysis, PDEs and/or Mathematical Physics. The topics are Nonlinear PDEs and variational methods, respectively, spectral and scattering properties of differential operators, in particular resonances.

 

New list of student projects (Mathematical analysis and numerical analysis)

Please visit Student infomation

 

Research Interests

Mathematical Physics, in particular the mathematics of Quantum Systems. Examples:

 

Nonlinear equations in quantum chemistry

Phase space bounds and eigenvalue asymptotics for Schrödinger operators

Spectral and scattering properties near thresholds for scalar-valued and matrix-valued Schrödinger operators
Resonances in Quantum Chemistry

Spectral and scattering properties of quantum wires

Spectral and scattering properties for Schrödinger and Dirac operators with a constant magnetic field

 

Scientific achievements

Here is a selection of my scientific achievements in the field of Mathematical Physics

(Numbers refer to the 'Refereed articles in journals' section of the publication list).

  • Rigorous justification of Complex Absorbing Potential method for perturbed Dirac operators in the semiclassical limit [KM.2014a,b]
  • Geometric properties of Stiefel and Grassmann manifolds in quantum chemistry [CM.2012d]
  • Existence of infinitely many distinct solutions to the quasi-relativistic multi-configurative Hartree-Fock equations [AM.2012a]
  • Rigorous results on perturbation of eigenvalues and half-bound states embedded at a threshold [M.2005a, M.2006].
  • First proofs of the diamagnetic inequality and the Lieb-Thirring inequality for the multi-vortex Aharonov-Bohm Hamiltonian [MOR.2004a].
  • Discovery and proof of the non-classical formula which governs the eigenvalue asymptotics near the Landau levels for weakly perturbed, even-dimensional Schrödinger and Dirac operators with constant magnetic fields [MR.2003b].
  • Asymptotic expansions in the low-energy limit of the resolvent and the scattering matrix associated with one-dimensional Schrödinger operators, where the potential belongs to an abstract class of short-range potentials, allowing certain non-local potentials [M.2002b]. These are the most detailed results since Faddeev's classic work on the scattering matrix.
  • First analytical proof (by means of a sort of diamagnetic inequality expressed via heat-kernels) of the celebrated Cwikel-Lieb-Rozenblum inequality for Schrödinger operators with external magnetic fields [MR.1996]. The inequality plays an important role in the proof of stability of matter within non-relativistic quantum mechanics.