Pretty good state transfer on double stars

Let A be the adjacency matrix of a graph $X$ and suppose U(t)=exp(itA). We view A as acting on $\cx^{V(X)}$ and take the standard basis of this space to be the vectors $e_u$ for $u$ in $V(X)$. Physicists say that we have perfect state transfer from vertex $u$ to $v$ at time $\tau$ if there is a scalar $\gamma$ such that $U(\tau)e_u = \gamma e_v$. (Since $U(t)$ is unitary, $\norm\gamma=1$.) For example, if $X$ is the $d$-cube and $u$ and $v$ are at distance $d$ then we have perfect state transfer from $u$ to $v$ at time $\pi/2$. Despite the existence of this nice family, it has become clear that perfect state transfer is rare. Hence we consider a relaxation: we say that we have pretty good state transfer from $u$ to $v$ if there is a complex number $\gamma$ and, for each positive real $\epsilon$ there is a time $t$ such that $\norm{U(t)e_u - \gamma e_v} < \epsilon$. Again we necessarily have $|\gamma|=1$. Godsil, Kirkland, Severini and Smith showed that we have have pretty good state transfer between the end vertices of the path $P_n$ if and only $n+1$ is a power of two, a prime, or twice a prime. (There is perfect state transfer between the end vertices only for $P_2$ and $P_3$.) It is something of a surprise that the occurrence of pretty good state transfer is characterized by a number-theoretic condition. In this paper we study double-star graphs, which are trees with two vertices of degree $k+1$ and all other vertices with degree one. We prove that there is never perfect state transfer between the two vertices of degree $k+1$, and that there is pretty good state transfer between them if and only if $4k+1$ is a perfect square.