Analog of Levinson's formula for a Schrödinger operator with long-range potential

Trace formulas of zeroth order are obtained for a radial Schrödinger operator with long-range potential $V(x)$ that decreases as $x\to\infty$ as the power $x^{-\alpha}$ with $1\leqslant\alpha\leqslant 2$. These formulas relate the increment of the phase shift in the continuum to the characteristics of the discrete spectrum and generalize Levinson's theorem to the case of slowly decreasing potentials.