Bound states asymptotics in the system with quantum wires in ℝ³

Abstract

In this paper we discuss spectral properties of the Schr"odinger operator acting in $L^2 (\R^3)$ with a delta potential localized on two infinite lines. The aim of analysis is to reveal how the configuration of lines affects the spectrum. In particular, we study the behaviour of spectral infimum with respect to $\theta$. The main result of the paper concerns the asymptotics of the function counting of discrete spectrum points for $\theta \to 0$, namely we show that the discrete spectrum points number below the threshold of the essential spectrum admits the asymptotics $\mathcal O (\theta^{-1})$.