Eigenvector degeneracy

To determine the degeneracy of the energy levels or, equivalently, of the eigenvalues of the number operator N, we must first obtain the eigenvectors 0z) for the ground state. These eigenvectors are determined by equation (4.29). When equation (4.18a) is substituted for a, equation (4.29) takes the form... 

To an energy eigenvalue or state of 3C, there will generally correspond several independent eigenvectors or state functions i,. . . , n is the degeneracy of the state. These functions must form a basis for a representation of the group G if is invariant under G. If i2 is an element of G... 

Thus the dispersion relation is symmetric about q = 0. It follows from eqs. (18) and (21) that the components of c ( q/) satisfy the same set of 3.v linear homogeneous equations as the components of the eigenvector c(q /). Therefore, if degeneracy is absent, e(q /) and c ( q /) can only differ by a phase factor (which preserves normalization). The physical properties of the system are independent of the choice of this phase factor, which we take to be... 

Each DCT(q) block appears ca times along the diagonal. The eigenvalues of DCT(q) are and their degeneracy is 1(a), the dimension of the IR a. This completes the solution to the problem of finding the frequencies and the eigenvectors of the dynamical matrix D(q), except for a consideration of extra degeneracies that may arise from time-reversal symmetry. 

Additional labels may be necessary to account for degeneracies of the eigenvectors i a . Optimization is now equivalent to solving Eq. (7.42). 1... 

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