Virtual Bound Levels in a Gap of the Essential Spectrum of the Weakly Perturbed Periodic Schrödinger Operator

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In the space L2(Rd)(d≤3) we consider the Schrödinger operator Hγ= - Δ + V(x) · + γW(x) · , where V(x) = V(x1, x2, ⋯ , xd) is a periodic function with respect to all the variables, γ is a small real coupling constant and the perturbation W(x) tends to zero sufficiently fast as | x| → ∞. We study so called virtual bound levels of the operator Hγ, i.e., those eigenvalues of Hγ which are born at the moment γ= 0 in a gap (λ-,λ+) of the spectrum of the unperturbed operator H0= - Δ + V(x) · from an edge of this gap while γ increases or decreases. We assume that the dispersion function of H0, branching from an edge of (λ-, λ+) , is non-degenerate in the Morse sense at its extremal set. For a definite perturbation (W(x) ≥ 0) we show that if d ≤ 2, then in the gap there exist virtual eigenvalues which are born from this edge. We investigate their number and an asymptotic behavior of them and of the corresponding eigenfunctions as γ→ 0. For an indefinite perturbation we estimate the multiplicity of virtual bound levels. In particular, we show that if d = 3 and both edges of the gap (λ-,λ+) are non-degenerate, then under additional conditions there is a threshold for the birth of the impurity spectrum in the gap, i.e., σ(Hγ)∩(λ-,λ+)=∅ for a small enough | γ|.

Original languageEnglish
Pages (from-to)307-345
Number of pages39
JournalIntegral Equations and Operator Theory
Issue number3
StatePublished - 1 Jul 2016

Bibliographical note

Publisher Copyright:
© 2016, Springer International Publishing.


  • Asymptotic behavior of virtual eigenvalues
  • Coupling constant
  • Perturbed periodic potential
  • Schrödinger operator
  • Virtual eigenvalues

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory


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