Workshop Nosevol #3

Soeren Fournais : Counter-examples to strong diamagnetism.

Consider a Schrödinger operator with magnetic field $B(x)$ in 2-dimensions. The classical diamagnetic inequality implies that the ground state energy, denoted by $lambda_1(B)$, with magnetic field is higher than the one without magnetic field. However, comparison of the ground state energies for different non-zero magnetic fields is known to be a difficult question. We consider the special case where the magnetic field has the form $b beta$, where $b$ is a (large) parameter and $beta(x)$ is a fixed function. One might hope that monotonicity for large field holds, i.e. that $lambda_1(b_1 beta) > lambda_1(b_2 beta)$ if $b_1>b_2$ are sufficiently large. We will display counterexamples to this hope and discuss applications to the theory of superconductivity in the Ginzburg-Landau model. This is joint work with Mikael Persson Sundqvist.