*Søren Fournais*

*ProfessorDepartment 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)* if*b1>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.