Partial Differential Equations

Areas of research interest of Daniel Elton

Pauli operators and zero modes operators such as the Schrödinger and Pauli operators corresponding to periodic electric and magnetic fields arise naturally in the study of the electronic properties of crystals. Such operators are also of independent mathematical interest, not least because they lie in the interesting gap between operators with short range potentials and those with confining potentials (e.g., the harmonic oscillator). The existence of a discrete symmetry group for periodic operators (the translation preserving the lattice of periods) can be used to show that the spectrum of these operators has a band-gap structure. Zero modes are simply the zero energy eigenfunctions of a given operator- e.g. the Pauli operator on 3 dimensional Euclidean space or the 3 sphere. Apart from their intrinsic interest, problems concerning zero modes have significant physical applications in areas such as the stability of matter and non-perturbative behaviour in quantum electrodynamics. The first example of zero modes were only found in 1986 and many of their properties are not yet understood.

• D. M. Elton, The local structure of zero mode producing magnetic potentials, Communications in Mathematcal Physics, 229 (2002) 121-139. CMP
• D. M. Elton, The Bethe-Sommerfeld conjecture for the three-dimensional periodic Landau operator, Reviews in Mathematical Physics, 16 (2004), no. 10, 1259-1290. RMP