Supersymmetry and Schr\"odinger-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 Schr\"odinger 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= (0 & A^*, A & 0) \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 Schr\"odinger 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 \(D\).