354 research outputs found

    The exponent in the orthogonality catastrophe for Fermi gases

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    We quantify the asymptotic vanishing of the ground-state overlap of two non-interacting Fermi gases in dd-dimensional Euclidean space in the thermodynamic limit. Given two one-particle Schr\"odinger operators in finite-volume which differ by a compactly supported bounded potential, we prove a power-law upper bound on the ground-state overlap of the corresponding non-interacting NN-particle systems. We interpret the decay exponent γ\gamma in terms of scattering theory and find γ=π2arcsinTE/2HS2\gamma = \pi^{-2}{\lVert\arcsin{\lvert T_E/2\rvert}\rVert}_{\mathrm{HS}}^2, where TET_E is the transition matrix at the Fermi energy EE. This exponent reduces to the one predicted by Anderson [Phys. Rev. 164, 352-359 (1967)] for the exact asymptotics in the special case of a repulsive point-like perturbation.Comment: Version as to appear in J. Spectr. Theory, References update

    A Szeg\H{o} Limit Theorem Related to the Hilbert Matrix

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    The Szeg\H{o} limit theorem by Fedele and Gebert for matrices of the type identity minus Hankel matrix is proved for the special case 1βπHN,α1-\frac{\beta}{\pi}H_{N,\alpha} where HN,αH_{N,\alpha} is the N×NN\times N-Hilbert matrix, α12\alpha\geq\frac{1}{2}, and βC\beta\in\mathbf{C}. The proof uses operator theoretic tools and a reduction to the classical Kac--Akhiezer theorem for the Carleman operator. Thereby, the validity of the theorem for this special Hankel matrix can be extended from β<1|\beta|<1 to βC]1,[\beta\in\mathbf{C}\setminus ]1,\infty[. The bound on the correction term is improved to O(1)O(1) instead of o(ln(N))o(\ln(N)) for βC[1,[\beta\in\mathbf{C}\setminus [1,\infty[. The limit case β=1\beta=1 is derived directly from the asymptotics for general β\beta
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