On the spectra of one-dimensional schrödinger operators with singular potentials

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Abstract

The paper is devoted to the spectral properties of one-dimensional Schrödinger operators Squ(x)=(-d2dx2+q(x))u(x), x∈ℝ, (1) with potentials q = qo+qs, where qo∈L∞(ℝ) is a regular potential, and qs∈D'(ℝ) is a singular potential with support on a discrete infinite set Y⊂ℝ. We consider the extension H of formal operator (1) to an unbounded operator in L2(ℝ) defined by the Schrödinger operator Sqo with regular potential qo and interaction conditions at the points of the set Y. We study the closedness and self-adjointness of H. If the set Y≃Z has a periodic structure we give the description of the essential spectrum of operator H in terms of limit operators. For periodic potentials qo we consider the Floquet theory of H, and apply the spectral parameter power series method for determining the band-gap structure of the spectrum. We also consider the case when the regular periodic part of the potential is perturbed by a slowly oscillating at infinity term. We show that this perturbation changes the structure of the spectra of periodic operators significantly. This works presents several numerical examples to demonstrate the effectiveness of our approach.

Original languageEnglish
Article number57
JournalFrontiers in Physics
Volume7
Issue numberMAR
DOIs
StatePublished - 2019

Keywords

  • Dispersion equation
  • Limit operators method
  • Monodromy matrices
  • Periodic Schrödinger operators
  • Slowly oscillating at infinity perturbation
  • Spectral parameter power series (SPPS) method

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