Spectral triangles of Schrödinger operators with complex potentials

• Plamen Djakov ^[1] ; Boris Mityagin ^[2]
  1. [1] Sofia University

     Sofia University

     Bulgaria

     
  2. [2] Ohio State University

     Ohio State University

     City of Columbus, Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 9, Nº. 4, 2003, págs. 495-528
• Idioma: inglés
• DOI: 10.1007/s00029-003-0358-y
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• Resumen

  • Consider the Schrödinger operator Ly=−y′′+v(x)y with a complex-valued potential v of period 1, v(x)=∑∞m=−∞V(2m)exp(2πimx). Let λ+n,λ−n and μn be the eigenvalues of L that are close to π2n2, respectively, with periodic (for n even), antiperiodic (for n odd), and Dirichelet boundary conditions on [0,1], and let Δn be the diameter of the spectral triangle with vertices λ+n,λ−n,μn. We prove the following statement: If ∑∞n=1Δ2n(1+n)2se2anb<∞,s≥0,a>0,b∈(0,1), then v(x) is a Gevrey function, and moreover ∑∞m=−∞|V(2m)|2(1+|m|)2se2a|m|b<∞.