Inverse spectral problem for singular Ablowitz–Kaup–Newell–Segur operators on [0, 1]

Abstract

Abstract. We consider an inverse spectral problem for a class of singular AKNS operators H a , a ∈ N with an explicit singularity. We construct for each a ∈ N, a standard map λ a × κ a with spectral data λ a and some norming constant κ a . For a = 0, λ a × κ a was known to be a local coordinate system on L 2Using adapted transformation operators, we extend this result to any non-negative integer a, give a description of isospectral sets and we obtain a Borg-Levinson type theorem.

Reconstruction of radial Dirac and Schrödinger operators from two spectra

“…For κ = 0, various aspects of the inverse theory for Dirac operators and systems have been developed by many authors; we point out, e.g., the pioneering paper [11] and recent articles [3,8,13,21,24] containing also extensive bibliography lists on the subject. Recently, the general case κ ∈ Z was considered in [5,25] for a related inverse spectral problem that uses one spectrum and corresponding norming constants to reconstruct the potential V . In [5] the class of potentials that belong to L p (0, 1), p ∈ [1, ∞), componentwise was treated and the double commutation method [12,14,27] was used, which allowed a reduction to the well-studied case κ = 0.…”

“…In [5] the class of potentials that belong to L p (0, 1), p ∈ [1, ∞), componentwise was treated and the double commutation method [12,14,27] was used, which allowed a reduction to the well-studied case κ = 0. In [25] a mapping between the spectral data and the potentials that are in L 2 (0, 1) componentwise was studied, paralleling the analysis of [23] for a regular Sturm-Liouville case and of [26] for the case of radial Schrödinger operators.…”

Inverse spectral problems for energy-dependent Sturm–Liouville equations

“…The corresponding norming constants α n (Q 0 ) are positive and the remaindersα n (Q 0 ) := α n (Q 0 ) − 1 form an ℓ 2 (Z)-sequence. Conversely, it follows from the results of [3,49] that every set {(λ n , α n )} n∈Z with λ n and α n possessing the above properties is the set of spectral data for a unique AKNS Dirac operator D(Q 0 ) with…”

Factorisation of Non-negative Fredholm Operators and Inverse Spectral Problems for Bessel Operators

“…We conclude with remark that similar results also hold for the Dirac operators with singular potentials appearing in the angular momentum decomposition of the radial Dirac operators in the unit ball of R 3 , cf. [2,22]. All constructions and proofs can be carried over by analogy with those presented here using the results of [2,22].…”

“…[2,22]. All constructions and proofs can be carried over by analogy with those presented here using the results of [2,22].…”