Expansion in characteristic functions of the Schrödinger operator with a singular potential

We study the spectral function of the operator $-\Delta+v(x)$ in three-dimensional space, where $v(x)$ is measurable and belongs to $L_2$. We study the differentiability of this function with respect to some measure. Simultaneously, we give estimates of the characteristic functions of a continuous spectrum at infinity. This justifies the decomposition of an arbitrary function in terms of the characteristic functions of an operator with this type of potential.