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Are there any study (theorems) on the eigenvalue gap (distribution of eigenvalues) of infinite dimensional Hermitian matrices? The general problem I'm considering here is:

Given an infinite series of $L\times L$ Hermitian matrices $\hat{H}_L,~L=1,2,3,\ldots$, let $E^i_L$ be the $i-$th eigenvalue of $\hat{H}_L$ (with $E^0_L\leq E^1_L\leq\ldots\leq E^{L-1}_L$), determine whether $$\Delta=\lim_{L\to\infty} (E^1_L-E^0_L)$$ is zero or a finite number.

A specific solvable example is $$H_{L,ij}=f(\frac{i}{L})\delta_{ij}-\frac{1}{L},$$ where $f(x)$ is a continuous function defined on $[0,1]$. It is easy to see that in the limit $L\to \infty$ all eigenvalues of $\hat{H}_L$ are distributed within $[\min\{f(x)\},\max\{f(x)\}]$ except the smallest one $E^0=\lim_{L\to \infty}E^0_L$ which is given by $$\int^1_0 \frac{dx}{f(x)-E^0}=1,$$ thus $E^0<\min\{f(x)\}$ and the spectrum is gapped $\Delta>0$.

I'm interested in whether there are any general results beyond this simple exactly solvable example. In general, it is assumed that the diagonal elements of $\hat{H}_L$ forms a stable, continuous distribution (called the "unperturbed spectrum") in the $L\to\infty$ limit, while the non-diagonal elements (called "perturbation" or "interaction") are either small (e.g. scales as $1/L$) or sparse.

This problem is of great importance in theoretical condensed matter physics, since whether a system has a gapped ($\Delta>0$) or gapless ($\Delta=0$) spectrum in the thermodynamic limit ($L\to\infty$) leads to fundamentally different macroscopic properties. Yet it seems to me that very little effort has been made to approach this problem on the mathematical side. Which research area in mathematics does this problem belong to? Ideas/references would be highly appreciated!

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