Time frequency localization in the Fourier Symmetric Sobolev space

We study concentration operators acting on the Fourier symmetric Sobolev space $H$ consisting of functions $f$ such that $\int_{\mathbb{R}} |f(x)|^2(1+x^2) dx + \int_{\mathbb{R}} |\hat{f}(\xi)|^2(1+\xi^2) d\xi < \infty$. We find that the Bargmann transform is a unitary operator from $H$ to a weighted Fock space. After identifying the reproducing kernel of $H$, we discover an unexpected phenomenon about the decay of the eigenvalues of a two-sided concentration operator, namely that the plunge region is of the same order of magnitude as the region where the eigenvalues are close to 1, contrasting the classical case of Paley--Wiener spaces.