Let $a>0$ be a fixed number and consider the Hermite operator (or harmonic oscillator) defined by
\begin{equation}
Hf(x)=x^2f(x)-f''(x)
\end{equation}
for any smooth function $f$ compactly supported on the interval $(-a,a)$. Then, we may extend $H$ as an unbounded operator on $L^2(-a,a)$.


Extending from the Wikipedia link : <https://en.wikipedia.org/wiki/Dirichlet_eigenvalue> I would like to compute the first Dirichlet eigenvalue $\lambda\_a$ of the above harmonic operator $H$ on the interval $(-a,a)$.


When working on whole $\mathbb{R}$, it is well-known that the first eigenvalue of $H$ is $1$. What I suspect from connection with the standard Gaussian measure is the following formula:
\begin{equation}
\lambda\_a=\frac{\sqrt{2\pi}}{\int\_{-a}^a e^{-\frac{x^2}{2}}dx}
\end{equation}


However, I cannot find a way to justify my guess. I tried to look for a relevant reference but all seem to deal with the Laplacian only, let alone the explicit formula for the first Dirichlet eigenvalue..


Could anyone please help me?



First Dirichlet eigenvalue of the harmonic oscillator on a bounded interval $(-a,a)$?
Your formula for the first Dirichlet eigenvalue cannot be correct: for $a=1/\sqrt{2}$, the first eigenvalue is $5$.
Indeed, the eigenfunction
$$y(x)=e^{-x^2/2}(2x^2-1)$$
is positive on $(-1/\sqrt{2},1/\sqrt{2})$, zero at the ends,
and satisfies
$$x^2y-y''=5y,$$
therefore $5$ is the smallest eigenvalue.
Your formula gives $2.071...$, which is not close.


For arbitrary $a$, eigenvalues and eigenfunctions can be expressed in terms of [Weber functions](https://en.wikipedia.org/wiki/Parabolic_cylinder_function).


