arXiv subjects

Martin Gebert

Publications and source records attributed to Martin Gebert.

At least 19 recordsLinked to original sources

arXiv2026

Szeg\H{o}-type determinant asymptotics for generalized Hilbert matrices with edge eigenvalues

We compute the large-$N$ asymptotics of the determinant of the identity plus or minus a generalized Hilbert matrix. For $N \in \mathbb{N}$ and $\delta \in \mathbb{R} \setminus \{-1, -2, \ldots\}$, let \[ H_N^\delta := \left( \frac{\sin((1+\delta)\pi)}{\pi(j+k+1+\delta)} \right)_{j,k=0}^{N-1} \] be the generalized Hilbert matrix. For $\delta$ not a half-integer, we prove the large-$N$ power-law asymptotics of the determinant \[ \log \text{det}(I_N \pm H_N^\delta) = -\frac{\theta_\delta^2 \pm \theta_\delta}{2} \log N + O(1) \] where $\theta_\delta := \delta$ if $\delta < -1/2$, and $\theta_\delta := \frac 1 \pi \arcsin(\sin(\delta\pi))$ if $\delta \ge -1/2$ and $\arcsin \colon [-1,1] \to [-\frac{\pi}{2}, \frac{\pi}{2}]$ denotes the principal branch of $\arcsin$. For $\delta \geq -\frac 12$ the asymptotics is known. The novelty of the present paper is the regime $\delta < - \frac 12$ and understanding the dichotomy in the decay exponent. This stems from the $\pm 1$ edge eigenvalues of the limiting operator appearing for $\delta < - \frac 12$. Most notably, we obtain for $\delta < -\frac12$ \[ \log \det\bigl(I_N - (H_N^\delta)^2\bigr) = -\delta^2 \log N + O(1). \] Such determinants arise in the study of Anderson's orthogonality catastrophe.

math-ph↗

arXiv2021

Lifshitz tails for random diagonal perturbations of Laurent matrices

We study the Integrated Density of States of one-dimensional random operators acting on $\ell^2(\mathbb Z)$ of the form $T + V_\omega$ where $T$ is a Laurent (also called bi-infinite Toeplitz) matrix and $V_\omega$ is an Anderson potential generated by i.i.d. random variables. We assume that the operator $T$ is associated to a bounded, H\"older-continuous symbol $f$, that attains its minimum at a finite number of points. We allow for $f$ to attain its minima algebraically. The resulting operator $T$ is long-range with weak (algebraic) off-diagonal decay. We prove that this operator exhibits Lifshitz tails at the lower edge of the spectrum with an exponent given by the Integrated Density of States of $T$ at the lower spectral edge. The proof relies on generalizations of Dirichlet-Neumann bracketing to the long-range setting and a generalization of Temple's inequality to degenerate ground state energies.

math-ph↗

arXiv2021

A Lieb-Robinson bound for quantum spin chains with strong on-site impurities

We consider a quantum spin chain with nearest neighbor interactions and sparsely distributed on-site impurities. We prove commutator bounds for its Heisenberg dynamics which incorporate the coupling strengths of the impurities. The impurities are assumed to satisfy a minimum spacing, and each impurity has a non-degenerate spectrum. Our results are proven in a broadly applicable setting, both in finite volume and in thermodynamic limit. We apply our results to improve Lieb-Robinson bounds for the Heisenberg spin chain with a random, sparse transverse field drawn from a heavy-tailed distribution.

math-ph↗

arXiv2021

On automatic extraction of on-street parking spaces using park-out events data

This article proposes two different approaches to automatically create a map for valid on-street car parking spaces. For this, we use car sharing park-out events data. The first one uses spatial aggregation and the second a machine learning algorithm. For the former, we chose rasterization and road sectioning; for the latter we chose decision trees. We compare the results of these approaches and discuss their advantages and disadvantages. Furthermore, we show our results for a neighborhood in the city of Berlin and report a classification accuracy of 91.6\% on the original imbalanced data. Finally, we discuss further work; from gathering more data over a longer period of time to fitting spatial Gaussian densities to the data and the usage of apps for manual validation and annotation of parking spaces to improve ground truth data.

cs.LG↗

arXiv2020

Dirichlet-Neumann bracketing for a class of banded Toeplitz matrices

We consider boundary conditions of self-adjoint banded Toeplitz matrices. We ask if boundary conditions exist for banded self-adjoint Toeplitz matrices which satisfy operator inequalities of Dirichlet-Neumann bracketing type. For a special class of banded Toeplitz matrices including integer powers of the discrete Laplacian we find such boundary conditions. Moreover, for this class we give a lower bound on the spectral gap above the lowest eigenvalue.

math.SP↗

arXiv2019

Lieb-Robinson bounds and strongly continuous dynamics for a class of many-body fermion systems in $\mathbb{R}^d$

We introduce a class of UV-regularized two-body interactions for fermions in $\mathbb{R}^d$ and prove a Lieb-Robinson estimate for the dynamics of this class of many-body systems. As a step toward this result, we also prove a propagation bound of Lieb-Robinson type for Schr\"odinger operators. We apply the propagation bound to prove the existence of infinite-volume dynamics as a strongly continuous group of automorphisms on the CAR algebra.

math-ph↗

arXiv2019

Lifshitz tails for the fractional Anderson model

We consider the $d$-dimensional fractional Anderson model $(-\Delta)^\alpha+ V_\omega$ on $\ell^2(\mathbb Z^d)$ where $0<\alpha\leq 1$. Here $-\Delta$ is the negative discrete Laplacian and $V_\omega$ is the random Anderson potential consisting of iid random variables. We prove that the model exhibits Lifshitz tails at the lower edge of the spectrum with exponent $ d/ (2\alpha)$. To do so, we show among other things that the non-diagonal matrix elements of the negative discrete fractional Laplacian are negative and satisfy the two-sided bound $$ \frac{c_{\alpha,d}}{|n-m|^{d+2\alpha}} \leq -(-\Delta)^\alpha(n,m)\leq \frac{C_{\alpha,d}}{|n-m|^{d+2\alpha}} $$ for positive constants $c_{\alpha,d}$, $C_{\alpha,d}$ and all $n\neq m\in\mathbb Z^d$.

math.PR↗

arXiv2019

On pure complex spectrum for truncations of random orthogonal matrices and Kac polynomials

Let $O(2n+\ell)$ be the group of orthogonal matrices of size $\left(2n+\ell\right)\times \left(2n+\ell\right)$ equipped with the probability distribution given by normalized Haar measure. We study the probability \begin{equation*} p_{2n}^{\left(\ell\right)} = \mathbb{P}\left[M_{2n} \, \mbox{has no real eigenvalues}\right], \end{equation*} where $M_{2n}$ is the $2n\times 2n$ left top minor of a $(2n+\ell)\times(2n+\ell)$ orthogonal matrix. We prove that this probability is given in terms of a determinant identity minus a weighted Hankel matrix of size $n\times n$ that depends on the truncation parameter $\ell$. For $\ell=1$ the matrix coincides with the Hilbert matrix and we prove \begin{equation*} p_{2n}^{\left(1\right)} \sim n^{-3/8}, \mbox{ when }n \to \infty. \end{equation*} We also discuss connections of the above to the persistence probability for random Kac polynomials.

math.PR↗

arXiv2018

On determinants identity minus Hankel matrix

In this note, we study the asymptotics of the determinant $\det(I_N - \beta H_N)$ for $N$ large, where $H_N$ is the $N\times N$ restriction of a Hankel matrix $H$ with finitely many jump discontinuities in its symbol satisfying $\|H\|\leq 1$. Moreover, we assume $\beta\in\mathbb C$ with $|\beta|<1$ and $I_N$ denotes the identity matrix. We determine the first order asymtoptics as $N\to\infty$ of such determinants and show that they exhibit power-like asymptotic behaviour, with exponent depending on the height of the jumps. For example, for the $N \times N$ truncation of the Hilbert matrix $\mathbf{H}$ with matrix elements $\pi^{-1}(j+k+1)^{-1}$, where $j,k\in \mathbb Z_+$ we obtain $$ \log \det(I_N - \beta \mathbf{H}_N) = -\frac{\log N}{2\pi^2} \big(\pi\arcsin(\beta)+\arcsin^2(\beta)+o(1)\big),\qquad N\to\infty. $$

math.FA↗

arXiv2018

A lower Wegner estimate and bounds on the spectral shift function for continuum random Schr\"odinger operators

We prove a strictly positive, locally uniform lower bound on the density of states (DOS) of continuum random Schr\"odinger operators on the entire spectrum, i.e. we show that the DOS does not have a zero within the spectrum. This follows from a lower Wegner estimate for finite-volume continuum random Schr\"odinger operators. We assume throughout iid random variables and the single-site distribution having a Lebesgue density bounded from below on its support. The main mathematical novelty in this paper are pointwise-in-energy bounds on the expectation of the spectral shift function at all energies for these operators where we mainly focus on perturbations corresponding to a change from Dirichlet to Neumann boundary conditions along the boundary of a cube. We show that the bound scales with the area of the hypersurface where the boundary conditions are changed. We also prove bounds on the averaged spectral shift function for perturbations by bounded and compactly supported multiplication operators.

math-ph↗

arXiv2017

On an integral formula for Fredholm determinants related to pairs of spectral projections

We consider Fredholm determinants of the form identity minus product of spectral projections corresponding to isolated parts of the spectrum of a pair of self-adjoint operators. We show an identity relating such determinants to an integral over the spectral shift function in the case of a rank-one perturbation. More precisely, we prove $$ -\ln \left(\det \big(\mathbf{1} -\mathbf{1} _{I}(A) \mathbf{1}_{\mathbb R\backslash I}(B)\mathbf{1}_{I}(A)\big) \right) = \int_I \text{d} x \int_{\mathbb R\backslash I} \text{d} y\, \frac{\xi(x)\xi(y)}{(y-x)^2}, $$ where $\mathbf{1}_J (\cdot)$ denotes the spectral projection of a self-adjoint operator on a set $J\in \text{Borel}(\mathbb R)$. The operators $A$ and $B$ are self-adjoint, bounded from below and differ by a rank-one perturbation and $\xi$ denotes the corresponding spectral shift function. The set $I$ is a union of intervals on the real line such that its boundary lies in the resolvent set of $A$ and $B$ and such that the spectral shift function vanishes there i.e. $I$ contains isolated parts of the spectrum of $A$ and $B$. We apply this formula to the subspace perturbation problem.

math.SP↗

arXiv2017

Perturbations of continuum random Schr\"odinger operators with applications to Anderson orthogonality and the spectral shift function

We study effects of a bounded and compactly supported perturbation on multi-dimensional continuum random Schr\"odinger operators in the region of complete localisation. Our main emphasis is on Anderson orthogonality for random Schr\"odinger operators. Among others, we prove that Anderson orthogonality does occur for Fermi energies in the region of complete localisation with a non-zero probability. This partially confirms recent non-rigorous findings [V. Khemani et al., Nature Phys. 11, 560-565 (2015)]. The spectral shift function plays an important role in our analysis of Anderson orthogonality. We identify it with the index of the corresponding pair of spectral projections and explore the consequences thereof. All our results rely on the main technical estimate of this paper which guarantees separate exponential decay of the disorder-averaged Schatten $p$-norm of $\chi_{a}(f(H) - f(H^{\tau})) \chi_{b}$ in $a$ and $b$. Here, $H^{\tau}$ is a perturbation of the random Schr\"odinger operator $H$, $\chi_{a}$ is the multiplication operator corresponding to the indicator function of a unit cube centred about $a\in\mathbb{R}^{d}$, and $f$ is in a suitable class of functions of bounded variation with distributional derivative supported in the region of complete localisation for $H$.

math-ph↗

arXiv2016

A bound on the averaged spectral shift function and a lower bound on the density of states for random Schr\"odinger operators on $\mathbb{R}^d$

We obtain a bound on the expectation of the spectral shift function for alloy-type random Schr\"odinger operators on $\mathbb{R}^d$ in the region of localisation, corresponding to a change from Dirichlet to Neumann boundary conditions along the boundary of a finite volume. The bound scales with the area of the surface where the boundary conditions are changed. As an application of our bound on the spectral shift function, we prove a reverse Wegner inequality for finite-volume Schr\"odinger operators in the region of localisation with a constant locally uniform in the energy. The application requires that the single-site distribution of the independent and identically distributed random variables has a Lebesgue density that is also bounded away from zero. The reverse Wegner inequality implies a strictly positive, locally uniform lower bound on the density of states for these continuum random Schr\"odinger operators.

math-ph↗

arXiv2016

On polynomial Lieb-Robinson bounds for the XY chain in a decaying random field

We consider the isotropic XY quantum spin chain in a random external field in the $z$ direction, with single site distributions given by i.i.d. random variables times the critical decaying envelope $j^{-1/2}$. Our motivation is the study of many-body localization. We investigate transport properties in terms of polynomial Lieb-Robinson (PLR) bounds. We prove a zero-velocity PLR bound for large disorder strength $\lambda$ and for small $\lambda$ we show a partial converse, which suggests the existence of non-trivial transport in the model.

math-ph↗

arXiv2015

The asymptotics of an eigenfunction-correlation determinant for Dirac-$\delta$ perturbations (Anderson's Orthogonality Catastrophe for Dirac-$\delta$)

We give a proof of the exact asymptotic behaviour in Anderson's Orthogonality Catastrophe for Dirac-$\delta$ perturbations. We prove the exact asymptotics of the scalar product of the ground states of two non-interacting Fermi gases confined to a $3$-dimensional ball $B_L$ of radius $L$ in the thermodynamic limit, where the underlying one-particle operators differ by a Dirac-$\delta$ perturbation. More precisely, we show the algebraic decay of the correlation determinant $\big|\det\big(\langle\varphi_j^L, \psi_k^L\rangle\big)_{j,k=1,...,N}\big|^2= L^{-\zeta(E)+ \text{o}(1)}$, as $N,L\to\infty$ and $N/|B_L|\to~ \rho>0$, where $\varphi_j^L$ and $\psi_k^L$ denote the lowest-energy eigenfunctions of the finite-volume one-particle Schr\"odinger operators. The decay exponent is given in terms of the s-wave scattering phase shift $\zeta(E):=\delta^2(\sqrt E)/{\pi^2}$. For an attractive Dirac-$\delta$ perturbation we conclude that the decay exponent $\frac 1 {\pi^2}\Vert\arcsin |T(E)/2|\Vert^2_{\text{HS}}$ found in [GKMO14] does not provide a sharp upper bound on the decay of the correlation determinant.

math-ph↗

arXiv2014

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 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 $N$-particle systems. We interpret the decay exponent $\gamma$ in terms of scattering theory and find $\gamma = \pi^{-2}{\lVert\arcsin{\lvert T_E/2\rvert}\rVert}_{\mathrm{HS}}^2$, 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↗

arXiv2014

Finite-size energy of non-interacting Fermi gases

We prove the asymptotics of the difference of the ground-state energies of two non-interacting $N$-particle Fermi gases on the half line of length $L$ in the thermodynamic limit up to order $1/L$. We are particularly interested in subdominant terms proportional to $1/L$, called finite-size energy. In the nineties Affleck and co-authors [Aff97, ZA97, AL94] claimed that the finite-size energy equals the decay exponent occuring in Anderson's orthogonality catastrophe. It turns out that the finite-size energy depends on the details of the thermodynamic limit and typically also includes a linear term in the scattering phase shift.

math-ph↗

arXiv2013

Anderson's orthogonality catastrophe

We give an upper bound on the modulus of the ground-state overlap of two non-interacting fermionic quantum systems with $N$ particles in a large but finite volume $L^d$ of $d$-dimensional Euclidean space. The underlying one-particle Hamiltonians of the two systems are standard Schr\"odinger operators that differ by a non-negative compactly supported scalar potential. In the thermodynamic limit, the bound exhibits an asymptotic power-law decay in the system size $L$, showing that the ground-state overlap vanishes for macroscopic systems. The decay exponent can be interpreted in terms of the total scattering cross section averaged over all incident directions. The result confirms and generalises P. W. Anderson's informal computation [Phys. Rev. Lett. 18, 1049--1051 (1967)].

math-ph↗