Eigenvalues upper bounds for the magnetic Schrödinger operator

We study the eigenvalues λk(HA,q) of the magnetic Schro ̈dinger operator HA,q associated with a magnetic potential A and a scalar potential q, on a compact Riemannian manifold M, with Neu- mann boundary conditions if ∂M ̸= ∅. We obtain various bounds on λ1(HA,q),λ2(HA,q) and, more generally on λk(HA,q). Some of them are sharp. Besides the dimension and the volume of the man- ifold, the geometric quantities which plays an important role in these estimates are: the first eigenvalue λ′′ (M) of the Hodge-de 1,1 Rham Laplacian acting on co-exact 1-forms, the mean value of the scalar potential q, the L2-norm of the magnetic field B = dA, and the distance, taken in L2, between the harmonic component of A and the subspace of all closed 1-forms whose cohomology class is in- tegral (that is, having integral flux around any loop). In particular, this distance is zero when the first cohomology group H1(M,R) is trivial. Many other important estimates are obtained in terms of the conformal volume, the mean curvature and the genus (in dimension 2). Finally, we also obtain estimates for sum of eigen- values (in the spirit of Kro ̈ger estimates) and for the trace of the heat kernel.