The spectral gap and principle eigenfunction of the random conductance model in a line segment

In this paper, we study the spectral gap and principle eigenfunction of the random walk in the line segment $[1,N]$ with conductances $c^{(N)}(x,x+1{)}_{1\le x<N}$ where $c^{(N)}(x,x+1)>0$ is the rate of the random walk jumping from site $x$ to site $x+1$ and vice versa. Writing $r^{(N)}(x,x+1):=1\mathrm{/}{c}^{(N)}(x,x+1)$, under the assumption \begin{equation*} \limsup_{N\to \infty}, \frac{1}{N}\sup_{1< m \le N}, \left| \sum_{x=2}^m r^{(N)}(x-1, x)- (m-1) \right|;=;0,, \end{equation*} we prove that the spectral gap, denoted by ${\mathrm{g}\mathrm{a}\mathrm{p}}_N$, of the process satisfies ${\mathrm{g}\mathrm{a}\mathrm{p}}_N=(1+o(1)){\pi }^2\mathrm{/}{N}^2$ and the principle eigenfunction $g_N$ with $g_N(1)=1$ corresponding to the spectral gap is well approximated by $h_N(x):=\cos ((x-1\mathrm{/}2)\pi \mathrm{/}N)$.