Geodesic
Levinson formula for perturbed Hill operator
Teoretičeskaâ i matematičeskaâ fizika, Tome 62 (1985) no. 2, pp. 196-209

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For a one-dimensional perturbed Hill operator $H$ for which the impurity potential has a finite first moment, a “Levinson series” is obtained. This series of relationships generalizes the well-known Levinson formula to the case when there is a periodic potential. The “Levinson series” is an effective tool for investigating the discrete spectrum in gaps (forbidden bands). In particular, it is shown that in the case of a reflectionless impurity potential with finite second moment there are no eigenvalues of the operator $H$ in the distant gaps of the spectrum. 
@article{TMF_1985_62_2_a2,
     author = {N. E. Firsova},
     title = {Levinson formula for perturbed {Hill} operator},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {196--209},
     publisher = {mathdoc},
     volume = {62},
     number = {2},
     year = {1985},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1985_62_2_a2/}
}
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N. E. Firsova. Levinson formula for perturbed Hill operator. Teoretičeskaâ i matematičeskaâ fizika, Tome 62 (1985) no. 2, pp. 196-209. http://geodesic.mathdoc.fr/item/TMF_1985_62_2_a2/