The Schrödinger-Poisson eigenmatrix problem
The existence of at least one stationary solution of the Schrödinger-Poisson system (modeling some semiconductor devices) is proved. After some transformations the nonlinear system to be solved is as follows: \(m\in {\mathbf N}\); \({\mathbf x}\in Q_{L}\) where \(Q_{L}\) is a cube in \({\mathbb{R}}^{d}\), \(d=1,2,3\); \(\Phi ={\phi_{m}({\mathbf x})}_{m\in {\mathbf N}}\), \(\phi_{m}({\mathbf x})\) satisfy periodic boundary conditions and: \[ -{\textstyle \frac{1}{2}} \Delta \phi _{m}({\mathbf x}) + (V_{\Phi}({\mathbf x}) +\widetilde{V} ({\mathbf x})) \phi_{m}({\mathbf x})=\mu_{m} \phi_{m}({\mathbf x}), \] \[ -\Delta V_{\Phi}({\mathbf x}) =n_{\Phi }({\mathbf x}) -n_{D}({\mathbf x}),\quad n_{\Phi}({\mathbf x})=\sum_{m}\lambda_{m}(\mu_{m}) |\phi_{m}({\mathbf x})|^{2}. \] The data are \(n_{D}({\mathbf x})\) and the functions \(\lambda_{m}(t)\). The existence of a solution \(\Phi\) is proved by a fixed point argument. In addition, bounds and an iteration scheme for computing \(\mu_{m}\) are given.
stationary solution of the Schrödinger-Poisson system
scientific article; zbMATH DE number 944355