Non-Weyl Asymptotics for Resonances of Quantum Graphs

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Description:	Davies, B (Kings) Monday 26 July 2010, 09:00-10.00

Created:

2010-07-28 09:04

Collection:

Analysis on Graphs and its Applications

Publisher:

Isaac Newton Institute

Davies, B

Language:

eng (English)

Credits:

Author:	Davies, B
Producer:	Steve Greenham


Abstract:	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 ${\cal G}_0$.

Abstract:

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 ${\cal G}_0$.

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