J. Spectr. Theory 4, 715-768 (2014) [DOI: 10.4171/JST/84]

Supersymmetry and Schrödinger-Type Operators with Distributional Matrix-Valued Potentials

Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols, and Gerald Teschl

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 (D, H1, H2) of the form

0 A*
A 0
in   L2(ℝ)2m   and   H1 = A* A,    H2 = A A*   in L2(ℝ)m.
Here A= Im (d/dx) + φ in L2(ℝ)m, with a matrix-valued coefficient φ = φ* ∈ L1loc(ℝ)m × m, m ∈ ℕ, thus explicitly permitting distributional potential coefficients Vj in Hj, j=1,2, where
Hj = - Im d2/dx2 + Vj(x),   Vj(x) = φ(x)2 + (-1)j φ'(x),   j=1,2.
Upon developing Weyl-Titchmarsh theory for these generalized Schrödinger operators Hj, with (possibly, distributional) matrix-valued potentials Vj, we provide some spectral theoretic applications, including a derivation of the corresponding spectral representations for Hj, j=1,2. Finally, we derive a local Borg-Marchenko uniqueness theorem for Hj, j=1,2, by employing the underlying supersymmetric structure and reducing it to the known local Borg-Marchenko uniqueness theorem for D.

MSC2010: Primary 34B20, 34B24, 34L05; Secondary 34B27, 34L10, 34L40.
Keywords: Sturm-Liouville operators, distributional coefficients, Weyl-Titchmarsh theory, supersymmetry.

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