STRUCTURE OF THE SPECTRUM OF THE SCHRODINGER OPERATOR WITH MAGNETIC-FIELD IN A STRIP AND INFINITE-GAP POTENTIALS

The Sturm-Liouville operator H = d(2)/dx(2) + V(x + p) on an interval [a, b] with zero boundary conditions is considered; here V is a strict ly convex function of class C-2 On the real line R and p is a numerica l parameter. The dependence of the eigenvalues of H on p is studied. T he spectral analysis of the Schrodinger operator with magnetic field i n a strip with Dirichlet boundary conditions on the boundary of the st rip reduces to this problem. As a consequence of the main result the f ollowing theorem is obtained. Let V-1 be the restriction of V to the i nterval [a, b) and let u be the periodic extension of V-1 on the entir e asis (with period b-a). Then all the gaps in the spectrum of the Sch rodinger operator -d(2)/dx(2) + u(x) are non-trivial. Bibliography: 19 titles.