arXiv subjects

Peter Otte

Publications and source records attributed to Peter Otte.

7 recordsLinked to original sources

arXiv2022

A Szegő Limit Theorem Related to the Hilbert Matrix

The Szegő limit theorem by Fedele and Gebert for matrices of the type identity minus Hankel matrix is proved for the special case $1-\fracβπH_{N,α}$ where $H_{N,α}$ is the $N\times N$-Hilbert matrix, $α\geq\frac{1}{2}$, and $β\in\mathbf{C}$. 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 $|β|<1$ to $β\in\mathbf{C}\setminus ]1,\infty[$. The bound on the correction term is improved to $O(1)$ instead of $o(\ln(N))$ for $β\in\mathbf{C}\setminus [1,\infty[$. The limit case $β=1$ is derived directly from the asymptotics for general $β$.

math-ph↗

arXiv2017

Ground-state energy of one-dimensional free Fermi gases in the thermodynamic limit

We study the ground-state energy of one-dimensional, non-interacting fermions subject to an external potential in the thermodynamic limit. To this end, we fix some (Fermi) energy $ν>0$, confine fermions with total energy below $ν$ inside the interval $[-L,L]$ and study the shift of the ground-state energy due to the potential $V$ in the thermodynamic limit $L\to\infty$. We show that the difference $\mathcal{E}_L(ν)$ of the two ground-state energies with and without potential can be decomposed into a term of order one (leading to the Fumi-term) and a term of order $1/L$, which yields the so-called finite size energy. We compute both terms for all possible boundary conditions explicitly and express them through the scattering data of the one-particle Schrödinger operator $-Δ+V$ on $L^2(\mathbb R)$.

math-ph↗

arXiv2016

The exponent in the orthogonality catastrophe for Fermi gases

We quantify the asymptotic vanishing of the ground-state overlap of two non-interacting Fermi gases in $d$-dimensional Euclidean space in the thermodynamic limit. Given two one-particle Schrödinger 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 $N$-particle systems. We interpret the decay exponent $γ$ in terms of scattering theory and find $γ= π^{-2}{\lVert\arcsin{\lvert T_E/2\rvert}\rVert}_{\mathrm{HS}}^2$, where $T_E$ is the transition matrix at the Fermi energy $E$. 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.

math-ph↗

arXiv2015

Anderson's orthogonality catastrophe in one dimension induced by a magnetic field

According to Anderson's orthogonality catastrophe, the overlap of the $N$-particle ground states of a free Fermi gas with and without an (electric) potential decays in the thermodynamic limit. For the finite one-dimensional system various boundary conditions are employed. Unlike the usual setup the perturbation is introduced by a magnetic (vector) potential. Although such a magnetic field can be gauged away in one spatial dimension there is a significant and interesting effect on the overlap caused by the phases. We study the leading asymptotics of the overlap of the two ground states and the two-term asymptotics of the difference of the ground-state energies. In the case of periodic boundary conditions our main result on the overlap is based upon a well-known asymptotic expansion by Fisher and Hartwig on Toeplitz determinants with a discontinuous symbol. In the case of Dirichlet boundary conditions no such result is known to us and we only provide an upper bound on the overlap, presumably of the right asymptotic order.

math-ph↗

arXiv2013

Anderson's Orthogonality Catastrophe for One-dimensional Systems

We derive rigorously the leading asymptotics of the so-called Anderson integral in the thermodynamic limit for one-dimensional, non-relativistic, spin-less Fermi systems. The coefficient, $γ$, of the leading term is computed in terms of the S-matrix. This implies a lower and an upper bound on the exponent in Anderson's orthogonality catastrophe, $\tilde CN^{-\tildeγ}\leq \mathcal{D}_N\leq CN^{-γ}$ pertaining to the overlap, $\mathcal{D}_N$, of ground states of non-interacting fermions.

math-ph↗

arXiv2011

Convergence of Infinite Series in Bosonic Second Quantization

The functor of second quantization as well as quadratic creation and annihilation operators on the bosonic Fock space are defined through possibly infinite series. The domain of convergence is investigated by precise number operator estimates and the Banach-Steinhaus Theorem. Some fundamental properties of the limit operators are derived.

math-ph↗

arXiv2009

Boundedness properties of fermionic operators

The fermionic second quantization operator $dΓ(B)$ is shown to be bounded by a power $N^{s/2}$ of the number operator $N$ given that the operator $B$ belongs to the $r$-th von Neumann-Schatten class, $s=2(r-1)/r$. Conversely, number operator estimates for $dΓ(B)$ imply von Neumann-Schatten conditions on $B$. Quadratic creation and annihilation operators are treated as well.

math-ph↗