Langer, M., Pruckner, R., & Woracek, H. (2023). Canonical systems whose Weyl coefficients have dominating real part. Journal d’Analyse Mathématique, 1–40. https://doi.org/10.1007/s11854-023-0297-9
E101-01 - Forschungsbereich Analysis E058-02-3 - Fachgruppe Patent- und Lizenzmanagement
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Journal:
Journal d'Analyse Mathématique
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ISSN:
0021-7670
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Date (published):
2023
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Number of Pages:
40
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Publisher:
HEBREW UNIV MAGNES PRESS
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Peer reviewed:
Yes
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Keywords:
Canonical system; Weyl coefficient; high-energy behaviour; singular integral
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Abstract:
For a two-dimensional canonical system y′ (t) = zJH(t)y(t) on the half-line (0, ∞) whose Hamiltonian H is a.e. positive semi-definite, denote by qH its Weyl coefficient. De Branges’ inverse spectral theorem states that the assignment H ↦ qH is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions. The main result of the paper is a criterion when the singular integral of the spectral measure, i.e. Re qH(iy), dominates its Poisson integral Im qH(iy) for y → +∞. Two equivalent conditions characterising this situation are provided. The first one is analytic in nature, very simple, and explicit in terms of the primitive M of H. It merely depends on the relative size of the off-diagonal entries of M compared with the diagonal entries. The second condition is of geometric nature and technically more complicated. It involves the relative size of the off-diagonal entries of H, a measurement for oscillations of the diagonal of H, and a condition on the speed and smoothness of the rotation of H.
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Project title:
Order and type of canonical systems: P 30715-N35 (FWF - Österr. Wissenschaftsfonds)
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Project (external):
Austrian Science Fund (FWF) ; Russian foundation of basic research (RFBR)