Consider large tridiagonal matrix (where $a$ and $b$ are real numbers):

$$M = \begin{pmatrix} a^2 & b & 0 & 0 & \cdots \\ b & (a+1)^2 & b & 0 & \cdots & \\ 0 & b & (a+2)^2 & b & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{pmatrix}$$

What can be said about eigenvalues? Are analytic expressions known?

Or at least properties of eigenvalues?

(Note: a cross-post from mathstack)