Summary: The paper deals with the Sturm-Liouville operator generated on the finite interval \([0,\pi]\) by the differential expression \(-y''+q(x)y\), where \(q=u'\), \(u\in L_{\varkappa}[0,\pi]\) for some \(\varkappa\geq 2\), and arbitrary regular boundary conditions. Consider two such operators with different potentials but the same boundary conditions. We prove that the difference between spectral decompositions \(S_m^1(f)-S_m^2(f)\) of this operators tends to zero as \(m\to\infty\) for any \(f\in L_{\mu}[0,\pi]\) in the norm of the space \(L_{\nu}[0,\pi]\) if the indices satisfy the inequality \(1/\varkappa+1/\mu-1/\nu\leq 1\) (except for the case \(\varkappa=\nu=\infty, \mu=1\)). In particular, in the case of a square summable function \(u\) the uniform equiconvergence on the whole interval \([0,\pi]\) is proved for an arbitrary function \(f\in L_2[0,\pi]\)