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We analyze $F_\pi(Q^2)$ and $F_{P\gamma}(Q^2)$, $P=\pi,\eta,\eta'$, withinthe local-duality (LD) version of QCD sum rules, which allows one to obtainpredictions for hadron form factors in a broad range of momentum transfers. Toprobe the accuracy of this approximate method, we consider, in parallel to QCD,a potential model: in this case, the exact form factors may be calculated fromthe solutions of the Schr\"odinger equation and confronted with the resultsfrom the quantum-mechanical LD sum rule. On the basis of our quantum-mechanicalanalysis we conclude that the LD sum rule is expected to give reliablepredictions for $F_\pi(Q^2)$ and $F_{\pi\gamma}(Q^2)$ in the region $Q^2 \ge5-6$ GeV$^2$. Moreover, the accuracy of the method improves rather fast withgrowing $Q^2$ in this region. For the pion elastic form factor, the data atsmall $Q^2$ indicate that the LD limit may be reached already at relatively lowvalues of momentum transfers, $Q^2\approx 4-8$ GeV$^2$; we therefore concludethat large deviations from LD in the region $Q^2=20-50$ GeV$^2$ reported insome recent theoretical analyses seem unlikely. The data on the($\eta,\eta')\to\gamma\gamma^*$ form factors meet very well the expectationsfrom the LD model. Surprisingly, the {\sc BaBar} results for the$\pi^0\to\gamma\gamma^*$ form factor imply a violation of LD growing with $Q^2$even at $Q^2\approx 40$ GeV$^2$, at odds with the $\eta,\eta'$ case and theexperience from quantum mechanics.

Pion elastic and $(\pi^0,\eta,\eta')\to\gamma\gamma^*$ transition form factors in a broad range of momentum transfers