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= (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 $A=I_m(d/dx)+\varphi$ 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 [H_j = - I_m \frac{d^2}{dx^2} + V_j(x), \quad V_j(x) = ϕ(x)^2 + (-1)^{j} ϕ'(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 $D$.

36 pages

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