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I had this weird thought that should be not too difficult, but I can't seem to resolve. So, we know that $-\Delta$ is positive and self-adjoint on $H^2(\mathbb{R}^d)$, which implies that all of its eigenvalues have to be nonnegative real number(s).

However, as I try to think of what its eigenfunctions can be, I can't seem to find any in $H^2(\mathbb{R}^d)$. Simplest case in $d=1$ suggests that only possible eigenfunctions are $e^{i\lambda x}$, which can't even make it into $L^2(\mathbb{R})$. If I'm not mistaken, the eigenfunctions in other dimensions would also be $e^{i \overrightarrow{a} \cdot x}$ although I'm not sure if these are exhaustive, but they are also not even in $L^2(\mathbb{R}^d)$, let alone $H^2(\mathbb{R}^d)$.

So, are there ANY eigenfunctions of $-\Delta$ in $H^2(\mathbb{R}^d)$? Or is it the case that $\Delta$ does not have any eigenvalues, as the self-adjointness simply implies if it has any, they should be nonnegative.

Odyssey
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    The spectrum of $- \Delta$ on $\mathbb{R}^n$ is $[0,\infty)$, and it's all essential spectrum (no eigenvalues). Here is detailed discussion: https://math.stackexchange.com/questions/766479/what-is-spectrum-for-laplacian-in-mathbbrn – Hans Engler Jan 27 '23 at 15:37

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