Estimates for the Weyl coefficient of a two-dimensional canonical system
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arXiv
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| Format: | Preprint |
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2021
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| author | Langer, Matthias Pruckner, Raphael Woracek, Harald |
| author_facet | Langer, Matthias Pruckner, Raphael Woracek, Harald |
| contents | 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 $q_H$ its Weyl coefficient. De Branges' inverse spectral theorem states that the assignment $H\mapsto q_H$ is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions.
We give upper and lower bounds for $|q_H(z)|$ and $\operatorname{Im} q_H(z)$ when $z$ tends to $i\infty$ non-tangentially. These bounds depend on the Hamiltonian $H$ near the left endpoint $a$ and determine $|q_H(z)|$ up to universal multiplicative constants. We obtain that the growth of $|q_H(z)|$ 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 $\operatorname{Im} q_H(z)$ is fully determined).
We translate the asymptotic behaviour of $q_H$ to the behaviour of the spectral measure $μ_H$ 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 $\pm\infty$ w.r.t. a weight function. Moreover, we apply our results to Krein strings and Sturm--Liouville equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_07391 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Estimates for the Weyl coefficient of a two-dimensional canonical system Langer, Matthias Pruckner, Raphael Woracek, Harald Mathematical Physics Functional Analysis 34B20, 37J99, 34L40, 34L20 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 $q_H$ its Weyl coefficient. De Branges' inverse spectral theorem states that the assignment $H\mapsto q_H$ is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions. We give upper and lower bounds for $|q_H(z)|$ and $\operatorname{Im} q_H(z)$ when $z$ tends to $i\infty$ non-tangentially. These bounds depend on the Hamiltonian $H$ near the left endpoint $a$ and determine $|q_H(z)|$ up to universal multiplicative constants. We obtain that the growth of $|q_H(z)|$ 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 $\operatorname{Im} q_H(z)$ is fully determined). We translate the asymptotic behaviour of $q_H$ to the behaviour of the spectral measure $μ_H$ 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 $\pm\infty$ w.r.t. a weight function. Moreover, we apply our results to Krein strings and Sturm--Liouville equations. |
| title | Estimates for the Weyl coefficient of a two-dimensional canonical system |
| topic | Mathematical Physics Functional Analysis 34B20, 37J99, 34L40, 34L20 |
| url | https://arxiv.org/abs/2106.07391 |