7 research outputs found
Anomalous electron trapping by localized magnetic fields
We consider an electron with an anomalous magnetic moment g>2 confined to a
plane and interacting with a nonzero magnetic field B perpendicular to the
plane. We show that if B has compact support and the magnetic flux in the
natural units is F\ge 0, the corresponding Pauli Hamiltonian has at least 1+[F]
bound states, without making any assumptions about the field profile.
Furthermore, in the zero-flux case there is a pair of bound states with
opposite spin orientations. Using a Birman-Schwinger technique, we extend the
last claim to a weak rotationally symmetric field with B(r) = O(r^{-2-\delta})
correcting thus a recent result. Finally, we show that under mild regularity
assumptions the existence can be proved for non-symmetric fields with tails as
well.Comment: A LaTeX file, 12 pages; to appear in J. Phys. A: Math. Ge
Weakly coupled states on branching graphs
We consider a Schr\"odinger particle on a graph consisting of links
joined at a single point. Each link supports a real locally integrable
potential ; the self--adjointness is ensured by the type
boundary condition at the vertex. If all the links are semiinfinite and ideally
coupled, the potential decays as along each of them, is
non--repulsive in the mean and weak enough, the corresponding Schr\"odinger
operator has a single negative eigenvalue; we find its asymptotic behavior. We
also derive a bound on the number of bound states and explain how the
coupling constant may be interpreted in terms of a family of
squeezed potentials.Comment: LaTeX file, 7 pages, no figure