Geodesic
Trace formulas for periodic and antiperiodic Sturm--Liouville problems
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (1982), pp. 6-11

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We find some trace-formulas for the Sturm–Liouville operator and its powers with periodic and antiperiodic boundary conditions. 
@article{VMUMM_1982_3_a1,
     author = {A. Sh. Shaldanbaev},
     title = {Trace formulas for periodic and antiperiodic {Sturm--Liouville} problems},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {6--11},
     publisher = {mathdoc},
     number = {3},
     year = {1982},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_1982_3_a1/}
}
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A. Sh. Shaldanbaev. Trace formulas for periodic and antiperiodic Sturm--Liouville problems. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (1982), pp. 6-11. http://geodesic.mathdoc.fr/item/VMUMM_1982_3_a1/