Divergence of spectral decompositions of Hill operators with two exponential term potentials

2012/10/15 by Plamen Djakov, Djakov, Plamen, Boris Mityagin +1
Computer Science · Mathematics · Physics and Astronomy · #34L10 #34L40 #47E05 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.CA #math.FA #math.MP #math.SP #msc:34L10 #msc:34L40 #msc:47E05

paper · pdf · doi:10.48550/arxiv.1210.3907

arxiv created 2012/10/15 · openalex publication_date 2012/10/15 · arxiv updated 2012/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Hill operator Ly = - y^{′ ′} + v(x)y, 0 ≤ x ≤ π, subject to periodic or antiperiodic boundary conditions (bc) with potentials of the form v(x) = a e^-2irx + b e^2isx, a, b ≠ 0, r,s ∈ ℕ, r≠ s. It is shown that the system of root functions does not contain a basis in L² ([0,π], ℂ) if bc are periodic or if bc are antiperiodic and r, s are odd or r=1 and s ≥ 3.

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