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).
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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 $-A_n$. |
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