Eigenvalue bounds for radial magnetic bottles on the disk

Francoise Truc (1)

1 IF - Institut Fourier

We consider a Schrödinger operator H with a non-vanishing radial magnetic field B=dA and Dirichlet boundary conditions on the unit disk. We assume growth conditions on B near the boundary which guarantee in particular the compactness of the resolvent of this operator. Under some assumptions on an additional radial potential V the operator H + V has a discrete negative spectrum and we obtain an upper bound on the number of negative eigenvalues. As a consequence we get an upperbound of the number of eigenvalues of H smaller than any positive value, which involves the minimum of B and the square of the L^2 -norm of A(r)/r, where A(r) is the specific magnetic potential defined as the flux of the magnetic field through the disk of radius r centerde in the origin.

Mots clés: magnetic bottles; eigenvalue bounds; Schrödinger operator

Domaines: Physique mathématique [math-ph]

Francoise Truc. Eigenvalue bounds for radial magnetic bottles on the disk. Asymptotic Analysis, 2012, 76, pp.233-248. ⟨10.3233/ASY-2011-1077⟩. ⟨hal-00588164v3⟩