Building on work on Miura's transformation by Kappeler, Perry, Shubin, and Topalov, we develop a detailed spectral theoretic treatment of Schrödinger operators with matrix-valued potentials, with special emphasis on distributional potential coefficients. Our principal method relies on a supersymmetric (factorization) formalism underlying Miura's transformation, which intimately connects the triple of operators of the form
Here in , with a matrix-valued coefficient , , thus explicitly permitting distributional potential coefficients in , , where
Upon developing Weyl–Titchmarsh theory for these generalized Schrödinger operators , with (possibly, distributional) matrix-valued potentials , we provide some spectral theoretic applications, including a derivation of the corresponding spectral representations for , . Finally, we derive a local Borg–Marchenko uniqueness theorem for , , by employing the underlying supersymmetric structure and reducing it to the known local Borg–Marchenko uniqueness theorem for .
Cite this article
Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols, Gerald Teschl, Supersymmetry and Schrödinger-type operators with distributional matrix-valued potentials. J. Spectr. Theory 4 (2014), no. 4, pp. 715–768DOI 10.4171/JST/84