Supersymmetry and Schrödinger-type operators with distributional matrix-valued potentials

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 D = ( 0 A∗ A 0 ) in L(R) and H1 = A∗A, H2 = AA∗ in L(R). Here A = Im(d/dx) + φ in L2(R)m, with a matrix-valued coefficient φ = φ∗ ∈ Lloc(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)jφ′(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.