Department of Mathematical Sciences
University of Aarhus
Consider a Schrödinger operator with magnetic field B(x) in 2-dimensions. The classical diamagnetic inequality implies that the ground state energy, denoted by λ1(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 bβ, where b is a (large) parameter and β(x) is a fixed function. One might hope that monotonicity for large field holds, i.e. that λ1(b1β)>λ1(b2β) ifb1>b2 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.