If you denote $A_n$ your tri-diagonal matrix of order $n$, and $H_n(x):= \det(x+A_n)$, the sequence $H_n$ saisfies satisfies the two term two-term linear recurrence $H_{n+1}=xH_n - nH_{n-1}$ with initial conditions $H_0=1$ and $H_1=x$. Thus, they are the Hermite polynomials (here in the "probabilist's version"), and their zeros are the eigenvalues of $-A_n$. -A_n$(on which you can find everything in the literature). 1 If you denote$A_n$your tri-diagonal matrix of order$n$, and$H_n(x):= \det(x+A_n)$, the sequence$H_n$saisfies the two term linear recurrence$H_{n+1}=xH_n - nH_{n-1}$with initial conditions$H_0=1$and$H_1=x$. Thus, they are the Hermite polynomials (here in the "probabilist's version"), and their zeros are the eigenvalues of$-A_n\$.