Imbedded Singular Continuous Spectrum for Schrödinger Operators

2001/11/19 by Alexander Kiselev, A. Kiselev, Kiselev, A.
Mathematics · Physics and Astronomy · #34L25 #34L40 #81U05 #FOS: Mathematics #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.SP #msc:34L25 #msc:34L40 #msc:81U05

paper · pdf · doi:10.48550/arxiv.math/0111200

30 pages, 2 figures

arxiv created 2001/11/19 · openalex publication_date 2001/11/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct examples of potentials V(x) satisfying |V(x)| ≤ (h(x))/(1+x), where the function h(x) is growing arbitrarily slowly, such that the corresponding Schrödinger operator has imbedded singular continuous spectrum. This solves one of the fifteen "twenty-first century" problems for Schrödinger operators posed by Barry Simon. The construction also provides the first example of a Schrödinger operator for which Möller wave operators exist but are not asymptotically complete due to the presence of singular continuous spectrum.

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