Given a sequence $\lbrace A_n \rbrace_n$ of matrices of increasing dimension ($A_n \in M_{d_n}(\mathbb{C})$ with $d_{n+1} > d_n$), we say that the sequence is distributed with respect to the eigenvalues as a function $f \in L^1$ on a domain D if $$ \lim_{n \to \infty} \frac{1}{n} \sum_{j=1}^{n} F(\lambda_j (A_n)) = \frac{1}{\mu(D)} \int_D F (f(x)) dx \quad \forall F \in \mathcal{C}_C (\mathbb{C}) $$ and one writes $$ \lbrace A_n \rbrace_{n \in \mathbb{N}} \underset{\lambda}{\sim} (f, D) $$ By analogy, we say the same for singular values distribution.
(See for istance https://www.springer.com/gp/book/9783319536781, chapter 3).
Given two matrix sequences $\lbrace A_n \rbrace_{n \in \mathbb{N}}, \lbrace B_n \rbrace_{n \in \mathbb{N}}$ which share the same distribution, ie $$ \lbrace A_n \rbrace_{n \in \mathbb{N}}, \lbrace B_n \rbrace_{n \in \mathbb{N}} \underset{\lambda}{\sim} (f,D) $$ what can one infer about the eigenvalues of each sequence?
Do $\lambda_j(A_n)$ and $\lambda_j(B_n)$ share any property?
My guess is, since $\big\lbrace \lbrace \lambda_1(A_n), ..., \lambda_{d(A)_n}(A_n) \rbrace, \lbrace x_j \rbrace_{j=1}^{d(A)_n} \big\rbrace$ and $\big\lbrace \lbrace \lambda_1(B_n), ..., \lambda_{d(B)_n}(B_n) \rbrace, \lbrace x_k \rbrace_{k=1}^{d(B)_n} \big\rbrace$, where $\lbrace x_j \rbrace_{j=1}^{d(A)_n}$ and $\lbrace x_k \rbrace_{k=1}^{d(B)_n}$ are equispaced grids on $D$, form an approximate reconstruction of the hypersurface $\lbrace x, f(x)\> s.t. \> x \in D \rbrace$, then it the difference (in a sense to be determined) between $\lbrace \lambda_1(A_n), ..., \lambda_{d(A)_n}(A_n) \rbrace$ and $\lbrace \lambda_1(B_n), ..., \lambda_{d(B)_n}(B_n) \rbrace$ is supposed to be smaller and smaller for increasing values of $n$.