WIAS Preprint No. 494, (1999)

Spectral properties of k - p Schrödinger operators in one space dimension


  • Bandelow, Uwe
    ORCID: 0000-0003-3677-2347
  • Kaiser, Hans-Christoph
  • Koprucki, Thomas
    ORCID: 0000-0001-6235-9412
  • Rehberg, Joachim

2010 Mathematics Subject Classification

  • 34L15 81Q10 34A45 34A50 34L10 34L40 47A75 65L60 81Q15 81-04


  • k - p Schrödinger operators with discontinuous coefficients, spectrum, analytic operator family, eigenvalue curves, approximation, discretization, band structure in layered semiconductors


In the physics of layered semiconductor devices the k · p method in combination with the envelope-function approach is a well established tool for band structure calculations. We perform a rigorous mathematical analysis of spectral properties for the corresponding spatially one dimensional k · p Schrödinger operators; thereby regarding a wide class of such operators. This class covers many of the k · p operators prevalent in solid state physics. It includes k · p Schrödinger operators with piecewise constant coefficients which is a prerequisite for dealing with the important case of semiconductor heterostructures. We also introduce a regularization of the problem which gives rise to a consistent discretization of k · p operators with jumping coefficients and describe our toolbox KPLIB for the numerical treatment of k · p operators. In particular we address the question of persistence of a spectral gap over the wave vector range.

Appeared in

  • Num. Funct. Anal. Opti. (2000), Vol. 21, no. 3/4, pp. 379-410

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