Dynamical and spectral Dirac systems: response function and inverse problems

Abstract

We establish simple connections between response functions of the dynamical Dirac systems and A-amplitudes and Weyl functions of the spectral Dirac systems. Using these connections we propose a new and rigorous procedure to recover a general-type dynamical Dirac system from its response function as well as a procedure to construct explicit solutions of this problem.MSC(2010): 34B20, 35Q41, 35B30, 37D99, 70Q05.

Evolution of Weyl Functions and Initial-Boundary Value Problems

“…Since, according to [57,Section 3], u(x, z) (z ∈ C M ) is a Weyl solution of (6.19) (i.e., u ∈ L 2 2 (0, ∞)), we see that y = K Y is a Weyl solution of (2.1), (2. Since m 1 = m 2 , it is possible to introduce a slightly different class of Weyl functions (that is, Weyl functions ϕ H ) via the inequality:…”

“…Theorem 6.5 [57] Let r(t) be the response function of a dynamical Dirac system and assume that r(t) admits representation r(t) = −2iϑ * 2 e −itα ϑ 1 , where the n × n (n ∈ N) matrix α and the column vectors ϑ i ∈ C N (i = 1, 2) satisfy the identity α − α * = −i ϑ 1 + ϑ 2 ϑ 1 + ϑ 2 * .…”

“…Formulas (6.26) and (6.28) imply that r(t) = 2i s ′ (t). Moreover, it is shown in [57] that the response function r coincides with a so called A-amplitude from [32] (more precisely, with the analogue of A-amplitude for the corresponding spectral Dirac system). We note that in M.G.…”

“…However, there is also an independent procedure in [49] (see [57,58] as well) to recover v from ϕ H . Instead of the structured operators S l given by (2.15) and (2.13), convolution operators S l : are used for this purpose.…”

“…In this section we give several statements from our paper [57], which appeared in arXiv in 2015. Classical Dirac systems (2.1) are also called spectral Dirac systems and various new results on inverse problems for these systems appeared quite recently (see some references in Subsection 2.1).…”