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Weak convergence of spectral shift functions for one-dimensional Schrödinger operators. (English) Zbl 1263.47026

The authors study the manner in which spectral shift functions associated with self-adjoint one-dimensional Schrödinger operators on the finite interval \((0,R)\) converge in the infinite volume limit \(R \to \infty \) to the half-line spectral shift function. Relying on a Fredholm determinant approach combined with certain measure theoretic facts, the authors show that prior vague convergence results in the literature in the special case of Dirichlet boundary conditions extend to the notion of weak convergence and arbitrary separated self-adjoint boundary conditions at \(x = 0\) and \(x = R\).

MSC:

47B25 Linear symmetric and selfadjoint operators (unbounded)
34L05 General spectral theory of ordinary differential operators
34L25 Scattering theory, inverse scattering involving ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34B24 Sturm-Liouville theory
34B27 Green’s functions for ordinary differential equations
47E05 General theory of ordinary differential operators
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