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Harald Woracek: High-energy behaviour of Weyl coefficients

Time: Wed 2021-10-13 13.15 - 14.15

Location: Zoom 692 1892 7142

Participating: Harald Woracek (TU Wien)

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We consider two-dimensional canonical systems \(y'(t) = zJH(t)y(t)\) on an interval \((0,L)\) where \(J\colon={\tiny\begin{pmatrix} 0 & {-1}\\ 1 & 0\end{pmatrix}}\), \(z\in\mathbb C\), and where the Hamiltonian \(H\colon(0,L)\to\mathbb R^{2\times 2}\) is locally integrable on \([0,L)\) with \(H(t)\geq 0\), \({\rm tr} H(t)>0\), and \(\int_0^L {\rm tr} H(t)\,dt=\infty\). Such systems have an operator model, which consists of a Hilbert space \(L^2(H)\), a self-adjoint operator (or linear relation), and a boundary map.

Given a Hamiltonian \(H\), Weyl's nested discs method produces a function \(q_H\) called its Weyl coefficient. It is a Nevanlinna function, i.e., analytic in the open upper half-plane with \({\rm Im}\,q_H(z)\geq 0\) (or \(q_H\equiv\infty\)). The operator \(A_H\) has simple spectrum, and a spectral measure \(\sigma_H\) is obtained from the Herglotz integral representation of \(q_H\). This makes it possible to investigate the spectrum of a canoncial system via the analytic function \(q_H\).

The behaviour of the Weyl coefficient \(q_H(z)\) when \(z\) approaches \(+i\infty\) is often named its high-energy behaviour. By classical Abelian–Tauberian theorems, it corresponds to the behaviour of the spectral measure at \(\pm\infty\).

In this talk we discuss some direct and some inverse spectral theorems which relate the high-energy behaviour of \(q_H\) to properties of the Hamiltonian \(H\). Among them regularly varying asymptotics, radial cluster sets, and dominating real part.

This talk is based on the arXiv preprints 210607391v1, 210604167v1, 210810162v1, and some manuscripts in preparation.