Let $S=\mathbb Z_{\ge0}$ be the nonnegative integers, and let $T=\mathbb R_{\ge0}$ be the nonnegative real numbers. Next, let $\gamma\in(0,1),r\in(0,1),$ and let $Q=(Q_{n,m})_{n,m\in S}$ be such that for all integers $n\ge0$ and $k\ge1,$ $$Q(n,n+k)=(n+1)(\gamma r)^k,\qquad Q(n+k,n)=(n+1)r^k,\qquad Q(n,n)=-\sum_{m\in S:m\ne n}Q(n,m).$$

Now, is there a precise formula for the spectral gap, $\operatorname{gap}(Q),$ of $Q$? (Spectral gap in the sense of reversible continuous-time Markov chains, where we interpret $Q$ as an infinitesimal generator.)

My work: We can compare $Q$ with the matrix $\tilde Q$ defined by $$\tilde Q(n,n+1)=(n+1)\gamma r,\quad\tilde Q(n,n-1)=nr.$$ The spectral gap for $\tilde Q$ is known to be $r(1-\gamma)$ (Chen '04, Section 9.3, second example under heading "Examples 9.27"), and for all $n,m\in S$ such that $n\ne m$ we have $\tilde Q(n,m)\le Q(n,m),$ so that $\operatorname{gap}(\tilde Q)\le\operatorname{gap}(Q).$ Thus, $r(1-\gamma)\le\operatorname{gap}(Q).$ I do not know how to improve this bound or to obtain an upper bound.

(Note: This question is a simplification of a previous post of mine. In the notation of that post, we have $\lambda_k=(\gamma r)^k$ and $\mu_k=r^k.$)

*Chen, Mu-Fa*, **From Markov chains to non-equilibrium particle systems.**, River Edge, NJ: World Scientific (ISBN 981-238-811-7/hbk; 978-981-256-245-6/ebook). xii, 597 p. (2004). ZBL1078.60003.