1d Schrödinger Operator with Periodic plus Compactly Supported Potentials

We consider the 1D Schrödinger operator Hy = −y′′ + (p+ q)y with a periodic potential p plus compactly supported potential q on the real line. The spectrum of H consists of an absolutely continuous part plus a finite number of simple eigenvalues in each spectral gap γn 6= ∅, n > 0, where γ0 is unbounded gap. We prove the following results: 1) we determine the distribution of resonances in the disk with large radius, 2) a forbidden domain for the resonances is specified, 3) the asymptotics of eigenvalues and antibound states are determined, 4) if q0 = ∫ R qdx = 0, then roughly speaking in each nondegenerate gap γn for n large enough there are two eigenvalues and zero antibound state or zero eigenvalues and two antibound states, 5) if H has infinitely many gaps in the continuous spectrum, then for any sequence σ = (σ) 1 , σn ∈ {0, 2}, there exists a compactly supported potential q such that H has σn bound states and 2 − σn antibound states in each gap γn for n large enough. 6) For any q (with q0 = 0), σ = (σn) ∞ 1 , where σn ∈ {0, 2} and for any sequence δ = (δn)∞1 ∈ l, δn > 0 there exists a potential p ∈ L(0, 1) such that each gap length |γn| = δn, n > 1 and H has exactly σn eigenvalues and 2− σn antibound states in each gap γn 6= ∅ for n large enough.