Consider the following symmetric second order diffusion operator, defined, for $\phi \in \mathcal{C}^{2,1}_c\left(\mathbb{R}\times \mathbb{R}_+\right)$, by: $$L\phi := \lambda_1 \partial_{R_1}(R_1 \phi) - \lambda_2 \partial_{R_2}((\sigma^2 - R_2)\phi) + \frac{1}{2}\lambda_1^2 \partial^2_{R_1, R_1}(\sigma^2\phi)$$ Where $\lambda_1, \lambda_2 > 0$ and $\sigma: (R_1, R_2) \in \mathbb{R}\times \mathbb{R}_+ \mapsto \beta_0 + \beta_1 R_1 + \beta_2 \sqrt{R_2}$, s.t $\beta_0 >0$, $\beta_1 < 0$ and $\beta_2 \in (0, 1]$.
I'm trying to find an expansion to the solutions of the Kolmogorov PDE defined by $L$, i.e: $$\partial_t u(t, R) = L u(t, R), \quad (t, R) \in (0, T) \times \mathbb{R}\times \mathbb{R}_+ $$
The expansion is defined by : $$\forall (t, R) \in [0, T] \times \mathbb{R}\times \mathbb{R}_+: \ u(t, R) = \sum_{k = 0}^{+\infty}\alpha_k e^{\kappa_k t} \Psi_k(R)$$
where $(\Psi_k)_{k\in \mathbb{N}}$ is the family of eigenvectors of $L$ associated with eigenvalues $(\kappa_k)_{k\in \mathbb{N}}$ and $(\alpha_k) \in \mathbb{R}^{\mathbb{N}}$.
Does anyone know how can we determine such eigenvalues/vectors ?