Abstract
This paper deals with a singular (Weyl’s limit circle case) non-self-adjoint (dissipative) Dirac operator with eigenparameter dependent boundary condition and finite general transfer conditions. Using the equivalence between Lax-Phillips scattering matrix and Sz.-Nagy-Foiaş characteristic function, the completeness of the eigenfunctions and associated functions of this dissipative operator is discussed.
Highlights
Spectral analysis and expansion of eigenfunctions in the fields of differential operators are important parts in the theory of ordinary differential equation boundary value problems
When an atomic system is subjected to an external electromagnetic field or a mechanical system to an external force, these would result in the discontinuity of origin system
Such as in geophysical problems, the reflection of transverse waves at the bottom of the earth’s crust jumps phenomena due to high-speed ions colliding with atomic systems. ese reasons may cause the eigenfunctions in the equations describing the system to have discontinuities, that is to say, operators with transfer conditions [9, 24, 25]
Summary
Spectral analysis and expansion of eigenfunctions in the fields of differential operators are important parts in the theory of ordinary differential equation boundary value problems. Behrndt et al [32] investigated an alternative approach to the construction of the self-adjoint dilation of an m-dissipative operator as well as a connection between the characteristic function and scattering matrix. We investigate a class of dissipative discontinuous Dirac operators with two singular endpoints (in Weyl’s limit circle case), and one of boundary conditions is linearly dependent on the eigen parameter, and general transfer conditions are imposed on the discontinuous points. E arrangement of this paper is as follows: in Section 2, we transfer the considered problem to a maximal dissipative operator Ah. In Section 3, the self-adjoint dilation, incoming and outgoing spectral representations, functional model, and characteristic function of this dissipative operator are derived.
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