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Let $\;W\in C^3(\mathbb R^2;\mathbb R)\;$ and the unbounded operator $\;A:W^{2,2}(\mathbb R;\mathbb R^2) \to L^2(\mathbb R;\mathbb R^2)\;$ which is defined by:

$\;Ay=-y''+(D^2W(z)y)^T\;$

I want to show that $\;A\;$ is a self-adjoint operator.

My Attempt:

The only definition I am familiar with, is this one $\;\langle Ay,x \rangle=\langle y,Ax \rangle\;$. However in order to be valid here the domain of $\;A\;$ should be also $\;L^2(\mathbb R;\mathbb R^2)\;$ so I can take the inner product in $\;L^2\;$.

How should I proceed? Am I missing some Theorems?

I have seen before only operators defined between the same Hilbert space and hence this confuses me a lot. Any help would be valuable!

Thanks in advance!

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Unbounded operators cannot be defined on the whole space. This operator is defined on a (presumably dense) subset of the space. Its range is in the same Hilbert space. (Hermitian) symmetric unbounded operators are not necessarily self-adjoint. See :Distinguishing between symmetric, Hermitian and self-adjoint operators. As Qiaochu Yuan comments, physicists use "self-adjoint" to mean symmetric.

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