Domain assumptions in singular-potential virial arguments

For the formal one-dimensional operator $H=p^2/(2m)-a|x{|}^{-p}$ with $a>0$ and $p>2$, take a smooth normalized test function supported away from the origin. Its normalized dilation ${\psi }_{\lambda }(x)=\sqrt{\lambda }\psi (\lambda x)$ hasAs $\lambda \to \infty$, this tends to minus infinity. Thus the unregularized expression has fall to the centre for a supercritical inverse-power potential and cannot define a semibounded Hamiltonian operator on a domain containing these test functions.

If a putative bound state has separately finite $T,U$ and admits norm-preserving dilation variations inside its operator domain or energy domain, stationarity gives $2T+pU=0$, hence $E=(1-2/p)T>0$, incompatible with a negative-energy bound state below the zero continuum threshold. For a putative ground state, the second scaling derivative is $2(2-p)T<0$, also incompatible with a minimum. The argument's finite-energy and domain assumptions are essential: it proves absence of a stable ground state and excludes bound states to which this ordinary virial calculation applies. An unqualified assertion about all formal square-integrable solutions at the singular origin needs a specified self-adjoint realization; adding a short-distance cutoff changes the problem.