Norm resolvent convergence of singularly scaled Schrödinger operators and δ′-potentials

2011/08/31 by Yu. D. Golovaty, R. O. Hryniv · 2 citations
Mathematics · Physics and Astronomy · #Advanced Harmonic Analysis Research #Convergence (economics) #Differential Equations and Boundary Problems #Interpretation (philosophy) #Limit (mathematics) #Norm (philosophy) #Operator (biology) #Operator theory #Real line #Resolvent #Resolvent formalism #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.SP

paper · pdf · doi:10.1017/s0308210512000194

published as Proceedings of the Royal Society of Edinburgh, 143A, 791-816, 2013 · 30 pages, 2 figure; submitted to Proceedings of the Royal Society of Edinburgh

arxiv created 2012/03/26 · openalex publication_date 2013/07/17 · arxiv updated 2013/09/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

For a real-valued function V of the Faddeev–Marchenko class, we prove the norm-resolvent convergence, as ε → 0, of a family S ε of one-dimensional Schrödinger operators on the line of the form Under certain conditions, the functions ε −2 V ( x / ε ) converge in the sense of distributions as ε → 0 to δ ′ ( x ), and then the limit S 0 of S ε may be considered as a ‘physically motivated’ interpretation of the one-dimensional Schrödinger operator with potential δ ′.

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