  We consider a model for the distribution of a long homopolymer with a
zero-range potential at the origin in $\mathbb{R}^3$. The distribution can be
obtained as a limit of Gibbs distributions corresponding to properly normalized
potentials concentrated in small neighborhoods of the origin as the size of the
neighborhoods tends to zero. The distribution depends on the length $T$ of the
polymer and a parameter $\gamma$ that corresponds, roughly speaking, to the
difference between the inverse temperature in our model and the critical value
of the inverse temperature.
  At the critical point $\gamma_{cr} = 0$ the transition occurs from the
globular phase (positive recurrent behavior of the polymer, $\gamma > 0$) to
the extended phase (Brownian type behavior, $\gamma < 0$). The main result of
the paper is a detailed analysis of the behavior of the polymer when $\gamma$
is near $\gamma_{cr}$.
  Our approach is based on analyzing the semigroups generated by the
self-adjoint extensions $\mathcal{L}_\gamma$ of the Laplacian on
$C_0^\infty(\mathbb{R}^3 \setminus \{0\})$ parametrized by $\gamma$, which are
related to the distribution of the polymer. The main technical tool of the
paper is the explicit formula for the resolvent of the operator
$\mathcal{L}_\gamma$.
