Reconstruction of radial Dirac and Schrödinger operators from two spectra

Abstract

We solve the inverse problem of reconstructing radial Dirac and Schrödinger operators acting in the unit ball of R 3 from two spectra of their one-dimensional parts corresponding to a fixed nonzero angular momentum. We give a complete description of the spectral data, prove existence and uniqueness of solutions to the inverse problem, and present the reconstruction algorithm.

Inverse uniqueness results for one-dimensional weighted Dirac operators

“…We apply our findings to the case of radial Dirac operators which have attracted significant interest recently [1,2,3,45]. In particular, we establish a Borg-Marchenko [5,37] result for this case extending the results from [9].…”

Spectral expansion for the singular Dirac system with impulsive conditions

“…is the Kronecker delta. The functions ψ (1) ij (x, λ) (i, j = 1, 2) are continuous both with respect to x ∈ [a, c) and λ ∈ R. Thus, for any ε > 0 there exists a c 0 < k < c such that…”

“…Let g (x) = { g (1) (x), x ∈ Ω 1 g (2) (x), x ∈ Ω 2 , g ∈ H be a vector-valued function that is equal to zero outside the finite interval [−τ, c) ∪ (c, τ ] , where τ ≥ s. Thus, we obtain ∫ c −τ ( f (1) s (x) , g (1) (x) 2) s (x) , g (2) (x)…”

“…On the other hand, direct or inverse spectral problems for Dirac operators were studied in [1,4,6,7,12,14,19,23,34,36,39,40]. In [13], Hıra and Altınışık investigated asymptotic behavior of eigenvalues and eigenfunctions of the discontinuous Dirac system, which includes an eigenvalue parameter in a transmission condition.…”

“…Let f (x) = {f(1) (x), x ∈ Ω 1 f (2) (x), x ∈ Ω 2 , f ∈ H.Then the integrals∫ ∞ −∞ F i (λ) ψ j (x, λ) dµ ij (λ) (i, j = 1, 2)converge in H. Consequently, we havef (i (λ) ψ j (x, λ) dµ ij (λ) ,which is called the expansion theorem.Proof Take any function f s ∈ H and any positive number s, and setf s (x) = i (λ) ψ j (x, λ) dµ ij (λ) ,wheref s (x) = { f…”