First Dirichlet eigenvalue of the harmonic oscillator on a bounded interval $(-a,a)$?

Question

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?

Answer 1

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.