Theoretical analysis reveals how harmonic oscillator zeros determine state overcompleteness in rational von Neumann lattices, highlighting mathematical connections to theta functions.
A rational von Neumann lattice is defined as a lattice in phase space with the constants a and b in the x and p directions given by a ratio of integers. Zeros of harmonic oscillator functions in the kq representation on such lattices are found. It is shown that the number of zeros of the kq function determines the number of states by which a set on a von Neumann lattice is overcomplete. Interesting relations between theta functions are derived on the basis of their connection with the harmonic oscillator states in the kq representation.
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Boon et al. (1983) studied this question.
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