[For the entire collection see Zbl 0539.00010.]
This short communication reports on some recent work by the authors, and announces a more detailed account thereof. The main result is an extension of the classical fact that on \({\mathcal C}_ 0^{\infty}({\mathbb{R}}^ N\setminus \{0\})\), the operator \(-\Delta -c| x|^{-2}\) is semibounded (actually nonnegative) if and only if \(c\leq C^*(N)\equiv (N-2)^ 2/4\) and essentially selfadjoint if and only if \(c\leq C^*(N)-1.\)
The authors consider the Cauchy problem for the heat equation \((*)\quad \partial u/\partial t=\Delta u+c| x|^{-2}u+f(x,t)\) with \(f\in L^ 1({\mathbb{R}}^ N\times (0,T))\), \(f\geq 0\), and initial data \(u(0)=\mu\), a non-negative finite measure. For \(c\leq C^*(N)\), they provide sufficient conditions and closely related necessary conditions on f and \(\mu\) for that problem to have a nonnegative solution. Those conditions are basically the integrability of \(| x|^{-\alpha}\mu\) and \(| x|^{-\alpha}f\) near 0, where \(\alpha\) is the smallest root of \(\alpha (N-2-\alpha)=c\). Conversely, for \(c>C^*(N)\), they show that the Cauchy problem has no nonnegative solutions.
The proofs, of which only sketchy excerpts are given, make use of an approximate equation where the potential \(c| x|^{-2}\) is truncated as \(V_ n\equiv c Min(| x|^{-2},n^ 2)\), and of the representation of the integral kernel for \(\exp [t(\Delta +V_ n)]\) through the Feynman-Kac formula.