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)
Since $M_n(a,b)$ and $M_n(a,-b)$ have same real spectrum, we may assume that $b\geq 0$. Let $\lambda_n$ be the smallest eigenvalue of $M_n$. Since there exist hidden othgonal polynomials, the real sequence $(\lambda_n)_n$ is non-increasing.
Assume that $a\geq 0$. Note that $e_1^TM_ne_1=a^2$; then $\lambda_n\leq a^2$. Denote by $B_n$ the matrix $M_n$ with a zero diagonal (only the $b$'s remain). Then $M_n\geq B_n$ and $\lambda_n\geq \inf(spectrum(B_n))\geq -2b$. Finally the sequence $(\lambda_n)_n$ converges to $\lambda\in [-2b,a^2]$.
Note that , if $\dfrac{b}{a^2}$ is small enough, then $M_n\geq 0$ and $\lambda\approx a^2$. If $a$ is fixed and $b$ tends to $+\infty$, then $\lambda\rightarrow -2b$.
