Characterization of potential smoothness and Riesz basis property of the Hill-Scrödinger operator in terms of periodic, antiperiodic and Neumann spectra

2012/07/04 by Batal, Ahmet
#34L40 #47E05 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1207.0948

Abstract

The Hill operators Ly=-y"+v(x)y, considered with complex valued π-periodic potentials v and subject to periodic, antiperiodic or Neumann boundary conditions have discrete spectra. For sufficiently large n, close to n² there are two periodic (if n is even) or antiperiodic (if n is odd) eigenvalues λ_n^-, λ_n⁺ and one Neumann eigenvalue ν_n. We study the geometry of "the spectral triangle" with vertices (λ_n⁺,λ_n^-,ν_n), and show that the rate of decay of triangle size characterizes the potential smoothness. Moreover, it is proved, for v∈ L^p ([0,π]), p>1, that the set of periodic (antiperiodic) root functions contains a Riesz basis if and only if for even n (respectively, odd n) sup_{λn⁺≠ λn^-}\|λ_n⁺-ν_n|/|λ_n⁺-λ_n^-| \ < ∞.

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