Article
J. Spectr. Theory 4, 715768 (2014)
[DOI: 10.4171/JST/84]
Supersymmetry and SchrödingerType Operators with Distributional MatrixValued Potentials
Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols, and Gerald Teschl
Building on work on Miura's transformation by Kappeler, Perry, Shubin, and Topalov,
we develop a detailed spectral theoretic treatment of Schrödinger operators with matrixvalued
potentials, with special emphasis on distributional potential coefficients.
D=
in L^{2}(ℝ)^{2m} and H_{1} = A^{*} A,
H_{2} = A A^{*} in L^{2}(ℝ)^{m}.
Here A= I_{m} (d/dx) + φ in L^{2}(ℝ)^{m}, with a
matrixvalued coefficient φ = φ^{*} ∈ L^{1}_{loc}(ℝ)^{m × m}, m ∈ ℕ,
thus explicitly permitting distributional potential coefficients V_{j} in H_{j}, j=1,2,
where
H_{j} =  I_{m} d^{2}/dx^{2} + V_{j}(x), V_{j}(x) = φ(x)^{2} + (1)^{j} φ'(x), j=1,2.
Upon developing WeylTitchmarsh theory for these generalized Schrödinger operators H_{j},
with (possibly, distributional) matrixvalued 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 BorgMarchenko uniqueness theorem for H_{j},
j=1,2, by employing the underlying supersymmetric structure and reducing it to the known
local BorgMarchenko uniqueness theorem for D.
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

MSC2010: Primary 34B20, 34B24, 34L05; Secondary 34B27, 34L10, 34L40.
Keywords: SturmLiouville operators, distributional coefficients, WeylTitchmarsh theory, supersymmetry.
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