Inverse Problem for Periodic “Weighted” Operators

✍ Scribed by Evgeni Korotyaev

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 290 KB
• Volume: 170
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1999.3479

✦ Synopsis

Define the periodic weighted operator Ty=&\ &2 ( \ 2 y$)$ in L 2 (R, (x) 2 dx). Suppose a function \ # W 2 1 (RÂZ) is 1-periodic real positive, (0)=1, and let q=$Â\ # L 2 (0, 1). The spectrum of T consists of intervals

, n 1, be the Dirichlet eigenvalue of the equation & y"&2qy$=z 2 y, y(0)=y(1)=0 where m n >0. Introduce the Lyapunov function 2(z, q) for T and note that

Let .(x, z, q) be the solution of the equation &."&2q.$=z 2 ., z # C, satisfying .(0, z, q)=0, .$(0, z, q)=1. Introduce the vector

where |h n | is defined by the equation cosh |h n | =(&1) n 2(z n , q) 1 and coincides with the euclidien norm of the vector h n . Using nonlinear functional analysis in Hilbert space, we prove that the mapping h: q Ä h(q)=[h n ] 1 is a real analytic isomorphism. 2000 Academic Press n&1 * & n * + n , n 1. These intervals are separated by the gaps # n =(* & n , * + n ), with the length |# n | 0. If a gap # n is degenerate, i.e.