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](https://math.stackexchange.com/questions/1670816/large-tridiagonal-matrix-eigenvalues))



Eigenvalues of large tridiagonal matrix
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 $\lambda\rightarrow -2b$.


