When dealing with Schrödinger operators with non-locally-integrable potentials or with Schrö\-din\-ger operators on graphs, it is convenient to split the domain into intervals \(I_i\) \((i\in \Omega)\) on which the potential is locally integrable. Then consider the differential operator \(T\) on the direct sum Hilbert space \(\bigoplus_{i\in\Omega }L^2(I_i,\omega _i)\) of weighted Lebesgue spaces with weight \(\omega _i\). The operator \(T=\bigoplus_{i\in \Omega}T_i\) is the direct sum of self-adjoint realizations of quasi-differential operators given by Shin-Zettl matrices. Some results on spectral representations, cyclic vectors, and spectral multiplicities are presented and are illustrated by examples.