Appendix: The Generalized Cayley-Hamilton Theorem

The original Cayley-Hamilton Theorem states that every square matrix satisfies its own characteristic equation (cf. e.g. [12,13]). It immediately follows from the theorem that the second rank SO(3) Cartesian tensor $\mathsf{q}$ satisfies the equation

$$-\mathsf{q}^3+q_1\mathsf{q}^2-q_2\mathsf{q}+q_3=0,$$ (51)

where $q_1$ is the trace, $q_2$ is the sum of the principal subdeterminants, and $q_3$ is the determinant of $\mathsf{q}$. Hence, a second rank tensor field $\mathsf{Q}(\mathsf{q})$ being the power series of the tensor $\mathsf{q}$

$$\mathsf{Q}(\mathsf{q})=c_0\mathsf{1}+c_1\mathsf{q}+c_2\mathsf{q}^2+c_3\mathsf{q}^3+\dots$$ (52)

can be sumed up to the form:

$$\mathsf{Q}(\mathsf{q})=\rho_0(q_1,q_2,q_3)\mathsf{1}+\rho_1(q_1,q_2,q_3)\mathsf{q} +\rho_2(q_1,q_2,q_3)\mathsf{q}^2,$$ (53)

where $\rho_1$, $\rho_2$ and $\rho_3$ are functions of scalars $q_1$, $q_2$ and $q_3$. The power series (52) is an isotropic function of $\mathsf{q}$ because it does not contain any other tensor.[7] For a symmetric traceless $(q_1=0)$ tensor $\underline{\mathsf{Q}}$ (a quadrupole tensor) Eq. (53) takes the form:

$$\underline{\mathsf{Q}}(\mathsf{q})=\varrho_1(q_2,q_3)\underline{\mathsf{q}} +\varrho_2(q_2,q_3)\underline{\mathsf{q}^2}$$ (54)

with two functions $\varrho_1$ and $\varrho_2$ of the two scalars $q_2$ and $q_3$. Eq. (54) is a direct consequence of the Cayley-Hamilton Theorem and thus can be called the Cayley-Hamilton theorem for the quadrupole tensors. It can be generalized for the isotropic tensor field $Q^{(L)}(q^{(\lambda )},q^{(\lambda^{\prime})},\dots )$ with an arbitrary multipolarity $L$ being a function of one or a few spherical tensors $q^{(\lambda )}$, $q^{(\lambda^{\prime})},\ \dots$ of ranks $\lambda$, $\lambda^{\prime},\ \dots$, respectively. The Generalized Cayley-Hamilton Theorem has the following general form:

$$Q^{(L)}(q^{(\lambda )},q^{(\lambda^{\prime})},\dots )=\sum_{k=1}^... ...\prime},\dots )} R_k(q)T^{(L)}_k(q^{(\lambda )},q^{(\lambda^{\prime})},\dots ),$$ (55)

where $q$ stands for the set of independent scalars, $T^{(L)}_k$ are some definite fundamental tensors, all constructed from $q^{(\lambda )}$, $q^{(\lambda^{\prime})},\ \dots$, and $R_k$ are arbitrary scalar functions. The number $k(L,\lambda ,\lambda^{\prime},\dots )$ depends on the ranks of the all involved tensors. To find the number and the forms of the fundamental tensors in a general case may appear to be a difficult task. A systematic method of constructing them in the case of the tensor fields depending on one tensor are presented in Refs.[6,8] Eqs. (20)-(22) are examples of the GCH theorem for $\lambda =\lambda^{\prime}=1$ and $L=0,\ 1,\ 2$, whereas Eqs. (23)-(25) -- for $\lambda =1$ and $L=0,\ 1,\ 2$. In the nuclear collective model the GCH theorem was used for $\lambda =2$ and $\lambda =3$.[14,15]