Weyl系数具有正则变化渐近性质的典范系统
Canonical systems whose Weyl coefficients have regularly varying asymptotics
AI总结：
研究二维典范系统Weyl系数趋向$i\infty$的渐近性质，证明其正则变化当且仅当Hamiltonian原函数正则变化且非对角项不振荡，并揭示了系数与原函数行为的显式关系及在谱测度中的应用。
摘要：
对于区间$(0,L)$（其中$0<L\le\infty$）上Hamiltonian $H$几乎处处为半正定的二维典范系统$y'(t)=zJH(t)y(t)$，记$q_H$为其Weyl系数。De Branges逆谱定理指出，对应关系$H\mapsto q_H$是迹归一化Hamiltonian与Nevanlinna函数之间的双射。我们证明，$q_H$趋向$i\infty$的渐近性质的首项是某个正则变化函数的（复）倍数，当且仅当$H$的原函数$M$在$0$处正则或快速变化，且其非对角项不振荡过于剧烈。$q_H$趋向$i\infty$的渐近首项通过显式公式与$M$在$0$处的行为相关联。绝对值的增长速度仅取决于$M$的对角项，而首项系数的辐角对应于非对角项的相对大小。转化到谱测度$μ_H$和Hamiltonian $H$上，这意味着$H$的对角项决定了$μ_H$对称化分布函数的增长，其非对角项的相对大小和符号分布是$μ_H$不对称性的度量。结果被应用于Sturm-Liouville方程、Krein弦和广义不定弦，以证明相应Weyl系数渐近性质的类似刻画。
英文摘要：
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.
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