I am interested in finding the gap between the two least eigenvalues of $A(t)$, a Hermitian $N\times N$ sparse matrix whose diagonal elements are $a_it+b_i\,(1\leq i\leq N)$, and all off-diagonal non-zero elements are the same and equal to $ct+d$. Furthermore, $A(t)$ is a banded matrix, that is, $a_{ij}=0$ for $( 1\leq i\leq N) \& (j>i+k)$ and $(1\leq j\leq N) \& (i> j+k)$, with $k<N$ fixed. Since real-scaled $N$ is larger than $10^6$, determining the eigenvalues and then calculating the gap $(g)$ is not practical. Thus, even bounding the gap can be satisfactory.
However, interlacing theorem and Courant-Weyl inequalities provide some bounds for the eigenvalues, but it is not enough since they involve the most and the least eigenvalues, so the bounds are to some extent wide. Another note I should mention is that while I need to calculate $g$ for a range of values for $t$, so my main (and optimistic) goal is to derive $g$ in terms of $t$. However, calculation of the gap (or its bounds) for some fixed $t_0 \neq 0$ can also do some good.
Now my question is that, according to the structure of $A(t)$, can I calculate $g$ or derive any better bound for it? Is there any sharp inequality that can help?
Any other useful comment would be appreciated.
Edit: To be more illustrative, here is an example of $A(t)$ (with $N=8$): $$\begin{bmatrix} 3-t & t-1 & t-1 & 0 & t-1 & 0 & 0 & 0 \\ t-1 & 3 (t+1) & 0 & t-1 & 0 & t-1 & 0 & 0 \\ t-1 & 0 & 3 (t+1) & t-1 & 0 & 0 & t-1 & 0 \\ 0 & t-1 & t-1 & 3-3 t & 0 & 0 & 0 & t-1 \\ t-1 & 0 & 0 & 0 & 7 t+3 & t-1 & t-1 & 0 \\ 0 & t-1 & 0 & 0 & t-1 & 3-t & 0 & t-1 \\ 0 & 0 & t-1 & 0 & t-1 & 0 & t+3 & t-1 \\ 0 & 0 & 0 & t-1 & 0 & t-1 & t-1 & 5 t+3 \\ \end{bmatrix}$$
and the plot for $g(t)$ for this example with $-1\leq t\leq 1 $ is:
Edit: Here is a long shot of an almost real-scaled example of $A(t)$ (with $N=32768$). While all off-diagonal non-zero elements are the same, Indeed, diagonal elements still vary from others (and may be from each other) -- though this is not visible at this scale.