Scattering matrices and Weyl functions

Abstract

Abstract:For a scattering system {A Θ , A 0 } consisting of selfadjoint extensions A Θ and A 0 of a symmetric operator A with finite deficiency indices, the scattering matrix {S Θ (λ)} and a spectral shift function ξ Θ are calculated in terms of the Weyl function associated with the boundary triplet for A * and a simple proof of the Krein-Birman formula is given. The results are applied to singular Sturm-Liouville operators with scalar and matrix potentials, to Dirac operators and to Schrödinger operators with point interactions.

Wave operators, similarity and dynamics for a class of Schrödinger operators with generic non-mixed interface conditions in 1D

“…In the case of selfadjoint extensions of a symmetric operator, a relation between the scattering matrix and the Weyl function, associated with a boundary triple, have been established in [4], while extensions of results from [4] to certain non-selfadjoint situations (dissipative/accumulative) have been presented in, [3], [5]. A generalization to the case of Q θ1,θ2 (V) would represents a useful insight in the study of the scattering properties of the system {Q θ1,θ2 (V), Q 0,0 (V)}.…”

“…A generalization to the case of Q θ1,θ2 (V) would represents a useful insight in the study of the scattering properties of the system {Q θ1,θ2 (V), Q 0,0 (V)}. 4 Further perspectives: the regime of quantum wells in a semiclassical island.…”

“…A boundary triple C 4 , Γ 0 , Γ 1 for Q(V) is defined with two linear boundary maps Γ i=1,2 : D(Q(V)) → C 4 fulfilling, for any ψ, ϕ ∈ D(Q(V)), the equation 4) and such that the transformation (…”

Zigzag chain model and its spectrum

“…This can be done in several ways. For example, this can be done using the so-called boundary triplet approach (see, for example, [25,26]) or the Von Neumann's formulas can be used (see, for instance, [27,28]). In our case, the domain of the adjoint operator in our model (due to the positivity of the operator and, correspondingly, regularity of negative points at the real axis) can be written in such manner:…”

Waveguide bands for a system of macromolecules

“…To construct a self-adjoint extension, it is more convenient to deal with the corresponding restriction of the adjoint operator. There are several ways to describe extensions, e.g., boundary triplets method ( [8,9], von Neumann formulas ( [10]), Krein resolvent formula ( [11,12]). We will use here a variant of the second approach which allows one, in the case of semi-boundedness of the Hamiltonian, to present an element from the domain of the adjoint operator in the following form:…”