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Weak convergence of spectral shift functions for one‐dimensional Schrödinger operators
Author(s) -
Gesztesy Fritz,
Nichols Roger
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201100222
Subject(s) - mathematics , fredholm determinant , convergence (economics) , boundary value problem , mathematical analysis , schrödinger's cat , real line , limit (mathematics) , spectral theory , dirichlet boundary condition , function (biology) , interval (graph theory) , spectral theory of ordinary differential equations , pure mathematics , hilbert space , quasinormal operator , combinatorics , evolutionary biology , economics , biology , finite rank operator , banach space , economic growth
We 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 the half‐line spectral shift function. Relying on a Fredholm determinant approach combined with certain measure theoretic facts, we 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 .