We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded domain in two dimensions. When the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the domain. We furt
hermore prove a lower bound for the first magnetic Neumann eigenvalue in the case of constant field.
We study the quadratic form associated to the kinetic energy operator in the presence of an external magnetic field in d = 3. We show that if the radial component of the magnetic field does not vanish identically, then the classical lower bound given
by Hardy is improved by a non-negative potential term depending on properties of the magnetic field.
