Spectrum of the \overline{∂}-Neumann Laplacian on polydiscs

Fu, Siqi

doi:10.1090/S0002-9939-06-08656-4

Spectrum of the $\overline {\partial }$-Neumann Laplacian on polydiscs
HTML articles powered by AMS MathViewer

by Siqi Fu PDF

Proc. Amer. Math. Soc. 135 (2007), 725-730 Request permission

Abstract:

The spectrum of the $\overline {\partial }$-Neumann Laplacian on a polydisc in $\mathbb {C}^n$ is explicitly computed. The calculation exhibits that the spectrum consists of eigenvalues, some of which, in particular the smallest ones, are of infinite multiplicity.

References

So-Chin Chen and Mei-Chi Shaw, Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. MR 1800297, DOI 10.1090/amsip/019
E. B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, 1995. MR 1349825, DOI 10.1017/CBO9780511623721
G. B. Folland, The tangential Cauchy-Riemann complex on spheres, Trans. Amer. Math. Soc. 171 (1972), 83–133. MR 309156, DOI 10.1090/S0002-9947-1972-0309156-X
G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Annals of Mathematics Studies, No. 75, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0461588
Siqi Fu, Hearing pseudoconvexity with the Kohn Laplacian, Math. Ann. 331 (2005), no. 2, 475–485. MR 2115465, DOI 10.1007/s00208-004-0591-5
—, Hearing the type of a domain in $\mathbb {C}^2$ with the $\overline \partial$-Neumann Laplacian, preprint, arXiv:math.CV/0508475.
Siqi Fu and Emil J. Straube, Compactness in the $\overline \partial$-Neumann problem, Complex analysis and geometry (Columbus, OH, 1999) Ohio State Univ. Math. Res. Inst. Publ., vol. 9, de Gruyter, Berlin, 2001, pp. 141–160. MR 1912737
Steven G. Krantz, Compactness of the $\overline \partial$-Neumann operator, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1136–1138. MR 954995, DOI 10.1090/S0002-9939-1988-0954995-2
G. N. Watson, A treatise on Bessel Functions, 2nd edition, Cambridge, 1948.