To simplify things a little, I describe conditions under which the smallest eigenvalue is strictly positive. These can be adjusted to get equality to zero.

**Necessary and sufficient** conditions for positive definiteness of the tridiagonal matrix in question are described below.

**Definition (Chain Sequence).** A sequence $\lbrace x_k \rbrace_{k > 0}$ is a chain sequence if there exists another sequence $\lbrace y_k \rbrace_{k\ge 0}$ such that
\begin{equation*}
x_k = y_k(1-y_{k-1}),
\end{equation*}
where $y_0 \in [0,1)$ and $y_k \in (0,1)$ for $k > 0$.

By the Wall-Wetzel Theorem, your tridiagonal matrix is positive definite **if and only if**

\begin{equation*}
\left\lbrace \frac{1}{a_ka_{k+1}} \right\rbrace_{k=1}^{n-1}
\end{equation*}

is a chain sequence.

**Example.** In particular, if the entries of the matrix satisfy,

\begin{equation*}
0 < \frac{1}{a_ka_{k+1}} < \frac{1}{4\cos^2\left(\frac{\pi}{n+1}\right)},\quad k=1,\ldots,n-1,
\end{equation*}
then it is positive definite.

For additional information and details about this material, please see:

- M. Andelic, and C. M. Da Fonesca.
*Sufficient conditions for positive definiteness of tridiagonal matrices revisited*. (2010).