  Positive definiteness of the quadratic part of the action of the
Hawking-Turok instanton is investigated. The Euclidean quadratic action for
scalar perturbations is expressed in terms of a single gauge invariant quantity
$q$. The mode functions satisfy a Schr\"odinger type equation with a potential
$U$. It is shown that the potential $U$ tends to a positive constant at the
regular end of the instanton. The detailed shape of $U$ depends on the initial
data of the instanton, on parameters of the background scalar field potential
$V$ and on a positive integer, $p$, labeling different spherical harmonics. For
certain well behaved scalar field potentials it is proven analytically that for
$p>1$ quadratic action is non-negative. For the lowest $p=1$ (homogeneous)
harmonic numerical solution of the Schr\"odinger equation for different scalar
field potentials $V$ and different initial data show that in some cases the
potential $U$ is negative in the intermediate region. We investigated the
monotonously growing potentials and a potential with a false vacuum. For the
monotonous potentials no negative modes are found about the Hawking-Turok
instanton. For a potential with the false vacuum the HT instanton is shown to
have a negative mode for certain initial data.
