Equiconvergence of spectral decompositions for Sturm--Liouville operators with a distributional potential in scales of spaces

A. M. Savchuk ; I. V. Sadovnichaya

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 496 (2021), pp. 56-58

Résumé

We study the equiconvergence of spectral decompositions for two Sturm–Liouville operators on the interval $[0,\pi]$ 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 $q_j(x)=u'_j(x)$, $u_i\in L_\kappa[0,\pi]$ 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 $\dfrac1\kappa+\dfrac1\mu+\dfrac1\nu\leq1$, $\mu,\nu\in[1,\infty]$, except for the case $\kappa=\nu=\infty$, $\mu=1$.

Détail

Keywords: Sturm–Liouville operator, distributional potentials, equiconvergence of spectral decompositions.