  The aim of this paper is to employ a strategy known from fluid dynamics in
order to provide results for the linear heat equation $u_{t}-\Delta u-V(x)u=0$
in $\mathbb{R}^{n}$ with singular potentials. We show well-posedness of
solutions, without using Hardy inequality, in a framework based in the Fourier
transform, namely $PM^{k}$-spaces. For arbitrary data $u_{0}\in PM^{k}$, the
approach allows to compute an explicit smallness condition on $V$ for global
existence in the case of $V$ with finitely many inverse square singularities.
As a consequence, well-posedness of solutions is obtained for the case of the
monopolar potential $V(x)=\frac{\lambda}{\left|x\right|^{2}}$ with
$\left|\lambda\right|<\lambda_{\ast}=\frac{(n-2)^{2}}{4}$. This threshold value
is the same one obtained for the global well-posedness of $L^{2}$-solutions by
means of Hardy inequalities and energy estimates. Since there is no any
inclusion relation between $L^{2}$ and $PM^{k}$, our results indicate that
$\lambda_{\ast}$ is intrinsic of the PDE and independent of a particular
approach. We also analyze the long time behavior of solutions and show there
are infinitely many possible asymptotics characterized by the cells of a
disjoint partition of the initial data class $PM^{k}$.
