The leading term of the spectral asymptotics for the Kohn--Laplace operator in a bounded domain
Matematičeskie zametki, Tome 64 (1998) no. 4, pp. 493-505.

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We prove the Weyl asymptotic formula for the number of eigenvalues of the Kohn–Laplace operator on a Heisenberg group and write out the leading term of asymptotics. The method of study is based on estimates of the Green function for the Dirichlet problem for the corresponding parabolic operator and makes use of the classical Hardy–Littlewood Tauberian theorem.
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Yu. A. Alkhutov; V. V. Zhikov. The leading term of the spectral asymptotics for the Kohn--Laplace operator in a bounded domain. Matematičeskie zametki, Tome 64 (1998) no. 4, pp. 493-505. http://geodesic.mathdoc.fr/item/MZM_1998_64_4_a1/

[1] Weyl H., “Das asymptotishe Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)”, Math. Ann., 71 (1912), 441–449 | DOI | MR

[2] Hörmander L., “Hypoelliptic second order differential equations”, Acta Math., 119 (1967), 147–171 | DOI | MR | Zbl

[3] Carleman T., “Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes”, C. R. Congr. der Math. Scand. (Stockholm, 1934), Lund, Stockholm, 1935, 34–44

[4] Minakshisundaram S., Pleijel A., “Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds”, Canad. J. Math., 1 (1949), 242–256 | MR | Zbl

[5] Gårding L., “On the asymptotic properties of the spectral function belonging to a self-adjoint semi-bounded extension of an elliptic differential operator”, Fysiogr. Sällsk. Lund Förhdl., 24:21 (1954), 19–32

[6] Kostyuchenko A. G., “Asimptoticheskoe raspredelenie sobstvennykh znachenii ellipticheskikh operatorov”, Dokl. AN SSSR, 158:1 (1964), 41–44 | MR | Zbl

[7] Kostyuchenko A. G., “Asimptoticheskoe povedenie spektralnoi funktsii samosopryazhennykh ellipticheskikh operatorov”, Chetvertaya matem. shkola, Kiev, 1968, 42–117 | Zbl

[8] Tsikon Kh., Freze R., Kirsh V., Saimon B., Operatory Shredingera s prilozheniyami v kvantovoi mekhanike i globalnoi geometrii, Mir, M., 1990

[9] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, T. 2, Mir, M., 1984 | Zbl

[10] Berezin F. A., Shubin M. A., Uravnenie Shredingera, Izd-vo MGU, M., 1983

[11] Folland G. B., Stein E. M., “Estimates for the $\overline\partial_b$ complex and analysis on the Heisenberg group”, Comm. Pure Appl. Math., 27 (1974), 429–522 | DOI | MR | Zbl