The Spectral Expansion for a Nonself-adjoint Hill Operator with a Locally Integrable Potential

✍ Scribed by O.A. Veliev; M.Toppamuk Duman

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 125 KB
• Volume: 265
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.2001.7693

✦ Synopsis

We construct the spectral expansion for the one-dimensional Schrödinger operator

, where q x is a 1-periodic, Lebesgue integrable on [0,1], and complex-valued potential. We obtain the asymptotic formulas for the eigenfunctions and eigenvalues of the operator L t , for t = 0, π, generated by this operation in L 2 0 1 and the t-periodic boundary conditions. Using it, we prove that the eigenfunctions and associated functions of L t form a Riesz basis in L 2 0 1 for t = 0, π. Then we find the spectral expansion for the operator L.