Canonical systems whose Weyl coefficients have regularly varying asymptotics
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| Format: | Preprint |
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2022
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| _version_ | 1866915444454064128 |
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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 an interval $(0,L)$ with $0<L\le\infty$
whose Hamiltonian $H$ is a.e.\ positive semidefinite, denote by $q_H$ its Weyl coefficient.
De~Branges' inverse spectral theorem states that the assignment
$H\mapsto q_H$ is a bijection between trace-normalised Hamiltonians and Nevanlinna functions.
We prove that $q_H$ has an asymptotics towards $i\infty$ whose leading term
is some (complex) multiple of a regularly varying function
if and only if the primitive $M$ of $H$ is regularly or rapidly varying at $0$
and its off-diagonal entries do not oscillate too much.
The leading term in the asymptotics of $q_H$ towards $i\infty$ is related
to the behaviour of $M$ towards $0$ by explicit formulae.
The speed of growth in absolute value depends only on the diagonal entries of $M$,
while the argument of the leading coefficient corresponds to the relative size
of the off-diagonal entries.
Translated to the spectral measure $μ_H$ and the Hamiltonian $H$,
this means that the diagonal of $H$ determines the growth of the
symmetrised distribution function of $μ_H$, and the relative size and
sign distribution of its off-diagonal is a measure for the asymmetry of $μ_H$.
The results are applied to Sturm--Liouville equations, Krein strings and generalised indefinite strings
to prove similar characterisations for the asymptotics of the corresponding Weyl coefficients. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2201_01522 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Canonical systems whose Weyl coefficients have regularly varying asymptotics Langer, Matthias Pruckner, Raphael Woracek, Harald Spectral Theory 34B20 (Primary), 45Q05, 30D40, 34L20 (Secondary) For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on an interval $(0,L)$ with $0<L\le\infty$ whose Hamiltonian $H$ is a.e.\ positive semidefinite, denote by $q_H$ its Weyl coefficient. De~Branges' inverse spectral theorem states that the assignment $H\mapsto q_H$ is a bijection between trace-normalised Hamiltonians and Nevanlinna functions. We prove that $q_H$ has an asymptotics towards $i\infty$ whose leading term is some (complex) multiple of a regularly varying function if and only if the primitive $M$ of $H$ is regularly or rapidly varying at $0$ and its off-diagonal entries do not oscillate too much. The leading term in the asymptotics of $q_H$ towards $i\infty$ is related to the behaviour of $M$ towards $0$ by explicit formulae. The speed of growth in absolute value depends only on the diagonal entries of $M$, while the argument of the leading coefficient corresponds to the relative size of the off-diagonal entries. Translated to the spectral measure $μ_H$ and the Hamiltonian $H$, this means that the diagonal of $H$ determines the growth of the symmetrised distribution function of $μ_H$, and the relative size and sign distribution of its off-diagonal is a measure for the asymmetry of $μ_H$. The results are applied to Sturm--Liouville equations, Krein strings and generalised indefinite strings to prove similar characterisations for the asymptotics of the corresponding Weyl coefficients. |
| title | Canonical systems whose Weyl coefficients have regularly varying asymptotics |
| topic | Spectral Theory 34B20 (Primary), 45Q05, 30D40, 34L20 (Secondary) |
| url | https://arxiv.org/abs/2201.01522 |