By F. Oort
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Additional info for Algebraic geometry, Oslo 1970; proceedings
Some simple combinatorics enables us to derive explicit general formulae for the density of states of a multidimensional quantum oscillator with arbitrary resonance condition n : n : 2 : n ) (Sadovskii and Zhilinskii, 1995). We also take into account the symmetry requirements that should be imposed if the molecule possesses some non-trivial symmetry. For the harmonic oscillator (the zero-order approximation) this problem can be completely solved by purely group-theoretical techniques based on the Molien generating function.
The "rst terms of this expansion read g ( , )"1#4 #10 #(20# ) #(35#4 ) # (56#12 ) #(84#28 # ) #2 . I. Zhilinskin& / Physics Reports 341 (2001) 85}171 136 We can also expand the generating function in Eq. (81) only in one auxiliary variable . Such an expansion has the form g ( , )" c(N ; ) ,Q Q . (1! , N ) sub-polyads. Q P #2 #2 #3 1 # # #2 . (85) g ( , )" (1! ) (1! ) (1! ) Q We can equally give more detailed formulae for numbers of states of certain symmetry within the (N , N ) sub-polyad.
BG ) in the denominator: g( )" L # L #2# LR . (1! B )2(1! BQ ) (60) Some of n and d can be identical. In the case of a generating function for invariants one of G H n should be zero. The expansion of g( ) in Eq. I. 1. The regular part of this quasi-polynomial C(N)"a NQ\#a NQ\#2#a #oscillatory part (62) Q\ Q\ can be directly found from Eq. (60) by replacing the formal parameter by e\U and expanding the result in the Laurent series in w at w"0: e\UL #e\UL #2#e\ULR "b w\Q#b w\Q>#2#b w\#2 .
Algebraic geometry, Oslo 1970; proceedings by F. Oort