Estimates for the Weyl coefficient of a two-dimensional canonical system

Langer, Matthias and Pruckner, Raphael and Woracek, Harald (2023) Estimates for the Weyl coefficient of a two-dimensional canonical system. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. ISSN 0391-173X (In Press)

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Abstract

For a two-dimensional canonical system y'(t)=zJH(t)y(t) on some interval (a,b) whose Hamiltonian H is a.e. positive semi-definite and which is regular at a and in the limit point case at b, 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. We give upper and lower bounds for |qH(z)| and Im qH(z) when z tends to i∞ non-tangentially. These bounds depend on the Hamiltonian H near the left endpoint a and determine |qH(z)| up to universal multiplicative constants. We obtain that the growth of |qH(z)| is independent of the off-diagonal entries of H and depends monotonically on the diagonal entries in a natural way. The imaginary part is, in general, not fully determined by our bounds (in forthcoming work we shall prove that for "most" Hamiltonians also Im qH(z) is fully determined). We translate the asymptotic behaviour of qH to the behaviour of the spectral measure μH of H by means of Abelian–Tauberian results and obtain conditions for membership of growth classes defined by weighted integrability condition (Kac classes) or by boundedness of tails at ±∞ w.r.t. a weight function. Moreover, we apply our results to Krein strings and Sturm–Liouville equations.