Eigenvalue Problem Of Magnetic Schr(?)dinger Operator On Network

In this paper, we study the superconductivity of network and the related problem of magnetic Schr(o|¨)dinger operator, which closely related to the eigenvalue problem of Schr(o|¨)dinger operator.When a network is placed in a varying magnetic field, it will undergo phase transation from the superconduct state to normal state.First, we show the existence of minimizers of the Ginzburg-Landau functional on network. Then, we study the general properties of the first eigenvalue, mainly focus on the existence, multiplicity, and the dependence on both the magnetic field and on the topology of the network. Finally, we calculate the first eigenvalues for several important examples.