2014/03/12 by Plamen Djakov, Djakov, Plamen, Boris Mityagin +1
Computer Science · Mathematics · #34L10 #34L40 #47E05 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1403.2973
openalex publication_date 2014/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the Hill operator L(v) = - d2/dx2 + v(x) on [0,\π] with\nDirichlet, periodic or antiperiodic boundary conditions; then for large enough\nn close to n2 there are one Dirichlet eigenvalue \μn and two periodic\n(if n is even) or antiperiodic (if n is odd) eigenvalues \λn-, ,\n\λn+ (counted with multiplicity).\n We describe classes of complex potentials v(x)= \∑2\ℤ V(k)≠ikx in weighted spaces (defined in terms of the Fourier coefficients of\nv) such that the periodic (or antiperiodic) root function system of L(v) \ncontains a Riesz basis if and only if V(-2n)
asymp V(2n)
quad
textas\n
;
; n
in 2
mathbbN
;
; (
textor
; n
in 1+ 2
mathbbN),
;
; n
to\n
infty. For such potentials we prove that \λn+ - \λn- \∼ \±\n2\√(V(-2n)V(2n)) and
mun -
frac12(
lambdan+ +
lambdan-)
sim\n-
frac12 (V(-2n) + V(2n)).\n