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

Abstract

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, H_1, H_2) of the form D= \left(\begin{smallmatrix} 0 & A^* \\ A & 0 \end{smallmatrix}\right) \, \text{ in } \, L^2(\mathbb{R})^{2m} \, \text{ and } \, H_1 = A^* A, \;\; H_2 = A A^* \, \text{ in } L^2(\mathbb{R})^m. Here A= I_m (d/dx) + \phi in L^2(\mathbb{R})^m , with a matrix-valued coefficient \phi = \phi^* \in L^1_{\text{loc}}(\mathbb{R})^{m \times m} , m \in \mathbb{N} , thus explicitly permitting distributional potential coefficients V_j in H_j , j=1,2 , where H_j = - I_m \frac{d^2}{dx^2} + V_j(x), \quad V_j(x) = \phi(x)^2 + (-1)^{j} \phi'(x), \; j=1,2. Upon developing Weyl–Titchmarsh theory for these generalized Schrödinger operators H_j , with (possibly, distributional) matrix-valued potentials V_j , we provide some spectral theoretic applications, including a derivation of the corresponding spectral representations for H_j , j=1,2 . Finally, we derive a local Borg–Marchenko uniqueness theorem for H_j , j=1,2 , by employing the underlying supersymmetric structure and reducing it to the known local Borg–Marchenko uniqueness theorem for D .