Equiconvergence of Spectral Decompositions for Sturm–Liouville Operators with a Distributional Potential in Scales of Spaces

Abstract—

We study the equiconvergence of spectral decompositions for two Sturm–Liouville operators on the interval [0, π] generated by the differential expressions ${{l}_{1}}(y) = - y'' + {{q}_{1}}(x)y$ and ${{l}_{2}} = - y'' + {{q}_{2}}(x)y$ and the same Birkhoff-regular boundary conditions. The potentials are assumed to be singular in the sense that ${{q}_{j}}(x) = u_{j}^{'}(x)$, ${{u}_{j}} \in {{L}_{\kappa }}[0,\pi ]$ for some $\kappa \in [2,\infty ]$ (here, the derivatives are understood in the sense of distributions). It is proved that the equiconvergence in the metric of ${{L}_{\nu }}[0,\pi ]$ holds for any function $f \in {{L}_{\mu }}[0,\pi ]$ if $\frac{1}{\kappa } + \frac{1}{\mu } - \frac{1}{\nu } \leqslant 1$, $\mu ,\nu \in [1,\infty ]$, except for the case $\kappa = \nu = \infty$, $\mu = 1$.