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 strictly convex function of class $C^2$ on the real line $\mathbb R$ and $p$ is a numerical parameter. The dependence of the eigenvalues of $H$ on $p$ is studied. The spectral analysis of the Schrödinger operator with magnetic field in a strip with Dirichlet boundary conditions on the boundary of the strip reduces to this problem. As a consequence of the main result the following theorem is obtained. Let $V_1$ be the restriction of $V$ to the interval $[a,b)$ and let $u$ be the periodic extension of $V_1$ on the entire axis (with period $b-a$). Then all the gaps in the spectrum of the Schrödinger operator $-d^2/dx^2+u(x)$ are non-trivial.