Supersymmetry and Schrodinger-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 Schrodinger operators with matrix-valued potentials, with special emphasis on distributional potential coecients. 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 2 (R) 2m and H1 = A A; H2 = AA in L 2 (R) m : Here A = Im(d=dx) + in L2(R)m, with a matrix-valued coecient = 2 L1 (R) m m, m2 N, thus explicitly permitting distributional potential coecients Vj in Hj, j = 1; 2, where Hj = Im d 2 dx2 +Vj(x); Vj(x) = (x) 2 + ( 1) j 0 (x); j = 1; 2: Upon developing Weyl{Titchmarsh theory for these generalized Schrodinger operators Hj, with (possibly, distributional) matrix-valued potentials Vj, we provide some spectral theoretic applications, including a derivation of the cor- responding 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.