The spectrum of the complex Laplacian for N -invariant pseudo-Kahlerian structures on C(n)
Abstract
Penney has shown that the complex Laplacian will have a trivial discrete spectrum when viewed as an operator on $L\sp2$ functions on $C\sp{n}$, where $C\sp{n}$ is equipped with an $N$-invariant pseudo-Kahlerian structure, and $N$ is "nice". However Penney has shown that there is a natural complex line bundle associated with these structures. When the Laplacian operates on $L\sp2$ sections of this complex line bundle, Penney has demonstrated that either there is no discrete spectrum, or there is no continuous spectrum. Penney also produced a mechanism for generating such $N$-invariant structures by means of linear functionals on the free abelian algebra of n-generators. The conditions for existence of discrete spectra have already been determined for the one generator case. We derive an algorithm for determining conditions in a more general setting, provide a partial result in the two generator case, and offer a program in the symbolic manipulation language MAPLE for implementing the algorithm.
Degree
Ph.D.
Advisors
Penney, Purdue University.
Subject Area
Mathematics
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