a_{\pm} \pi_k^{R,I} + b_{\pm} \phi_k^{R,I}$, with $$\begin{aligned}
&&a_\pm^2  = {\beta \over 2} \left[ 1 \pm \sqrt{1 - 4 {k^2 +m^2 \over \eta^2} } \right], \\
&&b_\pm^2 =  {2 \beta \over \eta^2} { (k^2 +m^2)^2 / \left[ 1 \pm \sqrt{1 - 4 {k^2 +m^2 \over \eta^2} 
} \right] }.\end{aligned}$$ The solutions of Eq. (\[eigen2\]) then are $$\begin{aligned}
&&E_N^\pm = N {a_\pm^2 \eta \over \beta} = N {\eta \over 2} 
\left[ 1 \pm \sqrt{1 - 4 {k^2 +m^2 \over \eta^2} } \right] \label{Solunder} 
\\ &&F_N^\pm = H_N [X_\pm ]. \nonumber\end{aligned}$$

These solutions can be brought to products of Hermite polynomials in $\pi_k^{R,I}$ and $\phi_k^{R,I}$ alone through the use of the property: $$\begin{aligned}
H_N[x +y] = 2 ^{-N/2} \sum_{i=0}^N \left( \stackrel{i}{N} \right) H_{N-i} [x \sqrt{2}] H_i [y \sqrt{2}].\end{aligned}$$ This property allow us to see that the solutions $$\begin{aligned}
P_N = \left( F_N [X_+] e^{- E_N^+t}  + F_N [X_-] e^{- E_N^- t} \right) P_0
\label{solcomb}\end{aligned}$$ form a complete orthogonal basis.

A few properties of these solutions are worth pointing out. The coefficients $a_\pm, b_\pm$ and the eigenvalues $E_N^\pm$ can now be complex numbers. This happens for $4 (k^2 +m^2) > \eta^2$. In this case $X_+ = X_-^*$, $E_N^+ 
= (E_N^-)^*$ and the combination in 