skip to content

Non-Weyl Asymptotics for Resonances of Quantum Graphs

Monday 26th July 2010 - 09:00 to 10:00
INI Seminar Room 1
Consider a compact quantum graph ${\cal G}_0$ consisting of finitely many edges of finite length joined in some manner at certain vertices. Let ${\cal G}$ be obtained from ${\cal G}_0$ by attaching a finite number of semi-infinite leads to ${\cal G}_0$, possibly with more than one lead attached to some vertices.

Let $H_0$ ( resp. $H$) $=-\frac{{\rm d}^2}{{\rm d} x^2}$ acting in $L^2({\cal G}_0)$ ( resp. $L^2({\cal G})$ ) subject to continuity and Kirchhoff boundary conditions at each vertex. The spectrum of $H$ is $[0,\infty)$, but unlike the normal case for Schrödinger operators $H$ may possess many $L^2$ eigenvalues corresponding to eigenfunctions that have compact support. However some eigenvalues of $H_0$ turn into resonances of $H$, and when defining the resonance counting function \[ N(r)=\#\{ \mbox{ resonances $\lambda=k^2$ of $H$ such that $|k|<r$}\} \] one should regard eigenvalues of $H$ as special kinds of resonance.

One might hope that $N(r)$ obeys the same leading order asymptotics as $r\to\infty$ as in the case of ${\cal G}_0$, but this is not always the case. A Pushnitski and EBD have proved the following theorem, whose proof will be outlined in the lecture.

Theorem 1 The resonances of $H$ obey the Weyl asymptotic law if and only if the graph ${\cal G}$ does not have any balanced vertex. If there is a balanced vertex then one still has a Weyl law, but the effective volume is smaller than the volume of ${\cal G}_0$.

The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.
Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons