2009/04/18 by Evgeny Korotyaev, Korotyaev, Evgeny
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.0904.2871
openalex publication_date 2009/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \gn≠ \es, n≥ 0, where \g0 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 \gn 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 \s=(\s)1^\iy, \sn∈ \0,2\, there exists a compactly supported potential q such that H has \sn bound states and 2-\sn antibound states in each gap \gn for n large enough. 6) For any q (with q0=0), \s=(\sn)1^\iy, where \sn∈ \0,2\ and for any sequence \d=(\dn)1^\iy∈ ℓ2, \dn>0 there exists a potential p∈ L2(0,1) such that each gap length |\gn|=\dn, n≥ 1 and H has exactly \sn eigenvalues and 2-\sn antibound states in each gap \gn≠ \es for n large enough.