# Harald Woracek: High-energy behaviour of Weyl coefficients

Time: Wed 2021-10-13 13.15 - 14.15

Location:

Lecturer: Harald Woracek (TU Wien)

### Abstract

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.

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