Let $n_1$, $n_2$, and $d$ be positive integers tending to infinity such that $$ d/n_k \to \phi_k \in (0,\infty). $$ Let $X_1 \in \mathbb R^{n_1 \times d}$ and $X_2^{n_2 \times d}$ be independent random matrices with iid entries from $N(0,1)$. Let $S_k := X_k^\top X_k/n_k \in \mathbb R^d$ be the sample covariance matrix corresponding to $X_k$. Fix $\lambda > 0$, unit-vectors $u,v \in \mathbb R^d$ at angle $\theta$ (i.e $u^\top v = \cos\theta$), and define
$$ \alpha := v^\top S_1(S_1 + S_2 + \lambda I_d)^{-1} u. $$
Question. How to express $\alpha$ (at least implicitly) as a function of $\phi_1$, $\phi_2$, $\theta$, and $\lambda$?
My hope is that properties of Stieltjes transforms would apply to give a kind of convolution, but I'm not very family with the theory / tools.
N.B. I'm particularly interested in the case $\phi_1,\phi_2 \in (0,1)$ and $\lambda \to 0^+$.
Generalization. More generally, one could consider $\alpha := \mathrm{tr}(A S_1 (S_1 + S_2 + \lambda I_d)^{-1})$, where $A$ is a (sequence of) $d \times d$ matrix with well-defined limiting spectral density, for example. The case considered abouve corresponds to the case where $A$ is a rank-$1$ matrix $uv^\top$.
Observations. In particular, it is clear that for $\phi_2 \to \infty$, one has $\alpha = v^\top S_1(S_1+\lambda I_d)^{-1}u + o(1)$, for which classical RMT (MP theory) directly applies.
