An eigenvalue problem for the Schrödinger-Maxwell equations
The authors study the eigenvalue problem for the Schrödinger operator coupled with the electromagnetic field \(E\), \(H\). The authors do not assume that the electromagnetic field is assigned, then they study a system of equations whose unknowns are the wave function \(\psi(x,t)\) and the gauge potentials \(A(x,t)\), \(\varphi(x,t)\) related to \(E\), \(H\). They investigate the case in which \(A\) and \(\varphi\) do not depend on time \(t\) and \(\psi(x,t)= u(x)e^{iwt}\), \(u\) a real function and \(\omega\) a real number. The problem is reduced to studying the existence of a real number \(\omega\) and real functions \(u\), \(\psi\) satisfying the system \[ -\textstyle{{1\over 2}}\Delta u- \varphi u=\omega u,\quad \Delta\varphi= 4\pi u^2\quad\text{in }\Omega,\quad u(x)= 0,\quad \varphi(x)= g\quad\text{on }\partial\Omega,\tag{1} \] \(\| u\|_{L_2}= 1\). Here \(g\) is an assigned function and \(\partial\Omega\) is the boundary of an open bounded set \(\Omega\subset \mathbb{R}^3\). The authors prove that there exists a sequence \((\omega_n,u_n,\varphi_n)\) with \(\omega_n\in \mathbb{R}\), \(\omega_n\to\infty\) and \(u_n\), \(\varphi_n\) real functions, solving (1).
scientific article; zbMATH DE number 1270104