On spectral properties of the Schreier graphs of the Thompson group $F$

In this article we study spectral properties of the family of Schreier graphs associated to the action of the Thompson group $F$ on the interval [0,1]. In particular, we describe spectra of Laplace type operators associated to these Schreier graphs and calculate certain spectral measures associated to the Schreier graph $\Upsilon$ of the orbit of 1/2. As a byproduct we calculate the asymptotics of the return probabilities of the simple random walk on $\Upsilon$ starting at 1/2. In addition, given a Laplace type operator $L$ on a tree-like graph we study relations between the spectral measures of $L$ associated to delta functions of different vertices and the spectrum of $L$.