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

  • Jonathan Eckhardt

    Universität Wien, Austria
  • Fritz Gesztesy

    Baylor University, Waco, USA
  • Roger Nichols

    The University of Tennessee at Chattanooga, USA
  • Gerald Teschl

    Universität Wien, Austria


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,H1,H2)(D, H_1, H_2) of the form

D=(0AA0) in L2(R)2m and H1=AA,    H2=AA in L2(R)m.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=Im(d/dx)+ϕA= I_m (d/dx) + \phi in L2(R)mL^2(\mathbb{R})^m, with a matrix-valued coefficient ϕ=ϕLloc1(R)m×m\phi = \phi^* \in L^1_{\text{loc}}(\mathbb{R})^{m \times m}, mNm \in \mathbb{N}, thus explicitly permitting distributional potential coefficients VjV_j in HjH_j, j=1,2j=1,2, where

Hj=Imd2dx2+Vj(x),Vj(x)=ϕ(x)2+(1)jϕ(x),  j=1,2.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 HjH_j, with (possibly, distributional) matrix-valued potentials VjV_j, we provide some spectral theoretic applications, including a derivation of the corresponding spectral representations for HjH_j, j=1,2j=1,2. Finally, we derive a local Borg–Marchenko uniqueness theorem for HjH_j, j=1,2j=1,2, by employing the underlying supersymmetric structure and reducing it to the known local Borg–Marchenko uniqueness theorem for DD.

Cite this article

Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols, Gerald Teschl, Supersymmetry and Schrödinger-type operators with distributional matrix-valued potentials. J. Spectr. Theory 4 (2014), no. 4, pp. 715–768

DOI 10.4171/JST/84