Geodesic
Analog of Levinson's formula for a~Schr\"odinger operator with long-range potential
Teoretičeskaâ i matematičeskaâ fizika, Tome 68 (1986) no. 2, pp. 244-254

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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. 
@article{TMF_1986_68_2_a7,
     author = {A. A. Kvitsinskiy},
     title = {Analog of {Levinson's} formula for {a~Schr\"odinger} operator with long-range potential},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {244--254},
     publisher = {mathdoc},
     volume = {68},
     number = {2},
     year = {1986},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1986_68_2_a7/}
}
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A. A. Kvitsinskiy. Analog of Levinson's formula for a~Schr\"odinger operator with long-range potential. Teoretičeskaâ i matematičeskaâ fizika, Tome 68 (1986) no. 2, pp. 244-254. http://geodesic.mathdoc.fr/item/TMF_1986_68_2_a7/