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Skew-self-adjoint discrete and continuous Dirac-type systems: inverse problems and Borg–Marchenko theorems

Alexander Sakhnovich

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New formulae on the inverse problem for the continuous skew-self-adjoint Dirac-type system are obtained. For the discrete skew-self-adjoint Dirac-type system the solution of a general-type inverse spectral problem is also derived in terms of the Weyl functions. The description of the Weyl functions on the interval is given. Borg–Marchenko-type uniqueness theorems are also derived for both discrete and continuous non-self-adjoint systems.


PACS

02.30.Zz Inverse problems

02.30.Uu Integral transforms

02.10.Yn Matrix theory

03.65.-w Quantum mechanics

02.30.Fn Several complex variables and analytic spaces

02.30.Tb Operator theory

MSC

30G25 Discrete analytic functions

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other quantum-mechanical equations

47B25 Symmetric and selfadjoint operators (unbounded)

47A56 Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)

34B20 Weyl theory and its generalizations

34A55 Inverse problems

37K15 Integration of completely integrable systems by inverse spectral and scattering methods

Subjects

Mathematical physics

Quantum information and quantum mechanics

Dates

Issue 6 (December 2006)

Received 7 January 2006, in final form 19 September 2006

Published 20 October 2006



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