Generating functions for effective Hamiltonians via the symmetrized Hadamard product
Boris I Zhilinskií · Journal of Physics A Mathematical and Theoretical · 2008
We present a method which enables one to calculate generating functions counting the number of linearly independent tensor operators of different degrees which should be included in phenomenological effective Hamiltonians constructed from boson creation and annihilation operators for several degrees of freedom in the presence of resonances and symmetry. The method is based on the application of the Molien generating function technique and the Hadamard product of rational functions. The latter leads to the representation of the answer in a form of a rational function. The technique is illustrated by the example of effective Hamiltonians for vibrational polyads in a methane-type molecule, which is a dynamical system with nine degrees of freedom formed by one non-degenerate, one doubly degenerate and two triply degenerate modes in resonance 2:1:1:2:2:2:1:1:1.