I'm writing an algorithm which computes the eigenvalues $E$ of the Schrödinger equation with potential $V(x) = x^2$, ie the harmonic oscillator. The equation is defined as follows
$$ y''(x) = (E-x^2)y(x) \, \, \text{where } x \in [-10, 10] $$
together with some boundary conditions, $y(-10) = y(10) = 0$. But how can I know if the output (the computed values $E$) is correct, or in other words, is there a list where I can find some of these $E$ values?