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

Here $A=I_m(d/dx)+\varphi$ in $L^2(\mathbb{R}{)}^m$, with a matrix-valued coefficient $\varphi ={\varphi }^{\ast }\in L_{\text{loc}}^1(\mathbb{R}{)}^{m\times m}$, $m\in \mathbb{N}$, thus explicitly permitting distributional potential coefficients $V_j$ in $H_j$, $j=1,2$, where

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 $D$.