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

Let $$a>0$$ be a fixed number and consider the Hermite operator (or harmonic oscillator) defined by $$$$Hf(x)=x^2f(x)-f''(x)$$$$ for any smooth function $$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: $$$$\lambda_a=\frac{\sqrt{2\pi}}{\int_{-a}^a e^{-\frac{x^2}{2}}dx}$$$$

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..

• What makes you think $H$ has eigenfunctions with compact support? Naively it looks like any functions with compact support are square integrable on $\mathbb{R}$, and all the square integrable eigenfunctions of $H$ are known. They are just the usual harmonic oscillator wave functions, none of which has compact support.
• I added more about the domain of $H$. Anyway, what I would like to figure out is the first Dirichlet eigenvalue of $H$ on the bounded interval $(-a,a)$, not the whole real line. Commented Jun 22, 2023 at 4:01
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.