Semiseparable Integral Operators and Explicit Solution of an Inverse Problem for a Skew-Self-Adjoint Dirac-Type System

Abstract

Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl matrix function is treated in the paper. Sufficient conditions under which the unique solution of the inverse problem exists, are formulated in terms of the Weyl function and a procedure to solve the inverse problem is given. The case of the generalized Weyl functions of the form φ(λ) exp{−2iλD}, where φ is a strictly proper rational matrix function and D = D * ≥ 0 is a diagonal matrix, is treated in greater detail. Explicit formulas for the inversion of the corresponding semiseparable integral operators and recovery of the Dirac-type system are obtained for this case.Mathematics Subject Classification (2010). Primary 34A55; Secondary 34B20; 47G10; 34A05.

Inverse problem for Dirac systems with locally square-summable potentials and rectangular Weyl functions

“…Moreover, it is easy to see that the proof of [7,Proposition 3.2] works also for the case, where ψ and ψ are differentiable functions with the squaresummable derivatives. Thus, recalling (3.9) and formulas (3.16) and (3.17) in [7,Proposition 3.2], we see that S ξ given by…”

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

“…If (5.41) holds, we can put in (5.42) M 1 = M . For a generalization of the Definition 5.9 see [16,23,54].…”

“…Thus, GBDT is a convenient tool to construct wave functions and explicit solutions of the nonlinear wave equations as well as to solve various direct and inverse problems. GBDT and its applications were treated or included as important examples in the papers [22,23,48,55,56,57,58,59,61,63,64,66,67,68,70,71,72,73,75] (see also [28,29,30,31,32,33,37]). Here we consider self-adjoint and skew-self-adjoint Dirac-type systems including the singular case corresponding to soliton-positon interaction and solve direct and inverse problems.…”

Operator identities corresponding to inverse problems for Dirac systems

“…The notation I will be used for the identity operator. The related operator identities, which appear in the case of skew-selfadjoint system (1.2), have the form (see [10,23] for the case that m 1 = m 2 )…”

“…To proceed with the proof we need Proposition 3.2 from [10], the formulation and proof of which are valid also for rectangular matrix functions k and k (though it is not stated in [10] directly). We rewrite Proposition 3.2:…”