Inverse spectral problem for singular AKNS and Schrödinger operators on[0,1]

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

“…Such operators were first introduced by Guillot and Ralston in [13] for the inverse spectral problem of the radial Schrödinger operator when a = 1; then used and extended to any integer a by Rundell and Sacks in [20] and by the present author in [21].…”

“…Next Zhornitskaya and Serov [24], and Carlson [7], proved that for all real a ≥ −1/2, λ a × κ a is one-to-one on L 2 R (0, 1). Finally, the author [21] completed theses works proving that for all a ∈ N the map λ a × κ a was a local (hence global) diffeomorphism on L 2 R (0, 1). Then, it is natural and interesting to wonder if these kind of results can be found for an other physical equation: the Dirac equation.…”

“…It may be seen as the analogue of the shift by a/2 in the eigenvalues asymptotics of the radial Schrödinger operator (see for instance [13] when a = 1 and [21] for general a).…”

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

“…The differential expression t(m; q) is well defined on the set of functions y that together with their quasiderivatives y [1] := y+(m/x)y are absolutely continuous on [ε, 1] for every ε ∈ (0, 1). It is well known [1,4,5,7,21] that being considered on the domain dom T (m; q) := {y ∈ dom t(m; q) ∩ L 2 (0, 1) | t(m; q)y ∈ L 2 (0, 1), y(1) = 0} the operator T (m; q) becomes self-adjoint, bounded below, and has a discrete spectrum. As earlier, for a nonzero λ ∈ C, we denote by y(·, λ) a solution of the equation…”

“…. in increasing order and recall [4,5,7,21] the asymptotic relation λ 2 n = π 2 n − m 2 2 + A +λ n , (4.11) IEOT with A ∈ R and an 2 -sequence (λ n ). Without loss of generality we assume that λ 2 1 > 0 as otherwise we can shift the spectrum by adding a suitable constant to q.…”