Geodesic
Asymptotics of the eigenvalues and the formula for the trace of perturbations of the Laplace operator on the sphere $\mathbb S^2$
Matematičeskie zametki, Tome 77 (2005) no. 3, pp. 434-448 
V. A. Sadovnichii; Z. Yu. Fazullin. Asymptotics of the eigenvalues and the formula for the trace of perturbations of the Laplace operator on the sphere $\mathbb S^2$. Matematičeskie zametki, Tome 77 (2005) no. 3, pp. 434-448. http://geodesic.mathdoc.fr/item/MZM_2005_77_3_a9/
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     title = {Asymptotics of the eigenvalues and the formula for the trace of perturbations of the {Laplace} operator on the sphere~$\mathbb S^2$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we study the asymptotics of the eigenvalues of the Laplace operator perturbed by an arbitrary bounded operator on the sphere $\mathbb S^2$. For the first time, for the partial differential operator of second order, the leading term of the second correction of perturbation theory is obtained. A connection between the coefficient of the second term of the asymptotics of the eigenvalues and the formula for the traces of the operator under consideration is established.

[1] Guillemin B., “Some spectral results for the Laplace operator with potential on the $n$-sphere”, Adv. in Math., 27 (1978), 273–286 | DOI | MR | Zbl

[2] Widom H., “The Laplace operator with potential on the $2$-sphere”, Adv. in Math., 31 (1979), 63–66 | DOI | Zbl

[3] Sadovnichii V. A., Dubrovskii V. V., “Klassicheskaya formula regulyarizovannogo sleda dlya sobstvennykh chisel operatora Laplasa–Beltrami s potentsialom na sfere”, Dokl. AN SSSR, 319:1 (1991), 61–62

[4] Podolskii V. E., “Formula regulyarizovannogo sleda operatora Laplasa–Beltrami s nechetnym potentsialom na sfere $\SS^2$”, Matem.zametki, 56:1 (1994), 71–77

[5] Fazullin Z. Yu., “Regulyarizovannyi sled operatora Laplasa–Beltrami”, Mezhdunarodnaya konferentsiya po kompleksnomu analizu i smezhnym voprosam, Tezisy dokl., Nizhnii Novgorod, 1997, 80–81

[6] Bobrov A. N., “Sled operatora Laplasa–Beltrami s potentsialom na poverkhnosti Tsollya”, Dokl. RAN, 368:2 (1999), 154–156 | Zbl

[7] Sadovnichii V. A., Fazullin Z. Yu., “Formula pervogo regulyarizovannogo sleda dlya vozmuscheniya operatora Laplasa–Beltrami”, Differents. uravneniya, 37:3 (2001), 402–409

[8] Sadovnichii V. A., Dubrovskii V. V., Poretskov O. A., “Formula pervogo regulyarizovannogo sleda operatora Laplasa–Beltrami s negladkim potentsialom na dvumernoi sfere”, Dokl. RAN, 382:1 (2002), 11–14 | Zbl

[9] Sadovnichii V. A., Fazullin Z. Yu., “Klasternaya asimptotika sobstvennykh chisel vozmuscheniya operatora Laplasa na sfere $\SS^2$”, Dokl. RAN, 391:4 (2003), 456–459

[10] Gobson E. V., Teoriya sfericheskikh i ellipsoidalnykh funktsii, IL, M., 1952

[11] Sege G., Ortogonalnye mnogochleny, GIFML, M., 1962

[12] Olver F., Asimptotika i spetsialnye funktsii, Nauka, M., 1990 | Zbl