I asked this question previously on math.stackexchange.com, where it had little traction.

Consider the symmetric random walk on $\{0,1,…,n\}$ with transition probabilities $P(j→j±1)=1/2$ for $0 < j < n$ and $P(0→0)=P(0→1)=P(n→n)=P(n→n−1)=1/2$. I am interested in the spectrum of the transition matrix (which is symmetric, hence the spectrum is real).

Mathematica suggests that the characteristic polynomial of the transition matrix is of the form $p_n(x)=(1−x)q_n(x)$, where $q_n$ is a polynomial that is odd / even iff $n−1$ is odd / even and that has only simple zeroes. Therefore, the spectrum appears to be a symmetric set of n points from the open unit interval, plus the point $λ=1$.

It occurs to me that this ought to be well known. In particular, the factors $q_n(x)$ in the characteristic polynomials ought to be special.

Does anybody know more?