Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The asymptotics of an eigenfunction-correlation determinant for Dirac-$\delta$ perturbations

Martin Gebert

Published Mar 12, 2015• 0 citations
OpenAlex

Abstract

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.

Related papers

The asymptotics of an eigenfunction-correlation determinant for Dirac-δ perturbations

Martin Gebert

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 BL of radius L in the thermodynamic limit, where the underlying one-particle operators differ by a Dirac-δ perturbation. More precisely, we show the algebraic decay of the correlation determinant det〈φjL,ψkL〉j,k=1,…,N2=L−ζ(E)+o(1), as N, L → ∞ and N/BL→ρ>0, where φjL and ψkL denote the lowest-energy eigenfunctions of the finite-volume one-particle Schrödinger operators. The decay exponent is given in terms of the s-wave scattering phase shift ζ(E)≔1π2δ2(E). For an attractive Dirac-δ perturbation, we conclude that the decay exponent 1π2arcsinT(E)/2HS2 found in the work of Gebert et al. [J. Spect. Theory (to be published); e-print arXiv:1407.2512] does not provide a sharp upper bound on the decay of the correlation determinant.

arXiv:1503.03654•Journal of Mathematical Physics•Jul 1, 2015•5 citations

Spectral and Eigenfunction correlations of finite-volume Schrödinger operators

Martin Gebert

The goal of this thesis is a mathematical understanding of a phenomenon called Anderson's Orthogonality in the physics literature. Given two non-interacting fermionic systems which differ by an exterior short-range scattering potential, the scalar product of the corresponding ground states is predicted to decay algebraically in the thermodynamic limit. This decay is referred to as Anderson's orthogonality catastrophe in the physics literature and goes back to P.W.Anderson [Phys. Rev. Lett. 18:1049--1051] and is used to explain anomalies in the X-ray absorption spectrum of metals. We call this scalar product $S_N^L$, where $N$ refers to the particle number and $L$ to the diameter of the considered fermionic system. This decay was proven in the works [Commun. Math. Phys. 329:979--998] and [arXiv:1407.2512] for rather general pairs of Schrödinger operators in arbitrary dimension $d\in\N$, i.e. $|S_N^L|^2\le L^{-\gamma}$ in the thermodynamic limit $N/L^d\to \rho>0$ approaching a positive particle density. In the general case, the biggest found decay exponent is given by $\gamma=\frac 1 {\pi^2} \norm{\arcsin|T/2|}_{\text{HS}}$, where T refers to the scattering T-matrix. In this thesis, we prove such upper bounds in more general situations than considered in both [Commun. Math. Phys. 329:979--998] and [arXiv:1407.2512]. Furthermore, we provide the first rigorous proof of the exact asymptotics Anderson predicted. We prove that in the $3$-dimensional toy model of a Dirac-$\delta$ perturbation that the exact decay exponent is given by $\zeta:= \delta^2/ \pi^2$. Here, $\delta$ refers to the s-wave scattering phase shift. In particular, this result shows that the previously found decay exponent $\gamma$ does not provide the correct asymptotics of $\S_N^L$ in general. Since the decay exponent is expressed in terms of scattering theory, these bounds depend on the existence of absolutely continuous spectrum of the underlying Schrödinger operators. We are able to deduce a different behavior in the contrary situation of Anderson localization. We prove the non-vanishing of the expectation value of the non-interacting many-body scalar product in the thermodynamic limit. Apart from the behavior of the scalar product of the non-interacting ground states, we also study the asymptotics of the difference of the ground-state energies. We show that this difference converges in the thermodynamic limit to the integral of the spectral-shift function up to the Fermi energy. Furthermore, we quantify the error for models on the half-axis and show that higher order error terms depend on the particular thermodynamic limit chosen.

•Jan 1, 2015•0 citations

The exponent in the orthogonality catastrophe for Fermi gases

Martin Gebert, Heinrich Küttler, Peter Müller +1 more

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 -Fermion systems. We interpret the decay exponent \gamma in terms of scattering theory and find \gamma = \pi^{-2}\|\mathrm {arcsin}|T_E/2|\|_{\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.

arXiv:1407.2512•Journal of Spectral Theory•Sep 15, 2016•9 citations

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

Peter Otte, Spitzer, Wolfgang

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 $L^2(\mathbb R)$.

arXiv:1801.00144•Dec 30, 2017•1 citation

Heat production of noninteracting fermions subjected to electric fields

J.-B. Bru, de Siqueira Pedra, W.

Efficiently bounding large determinants is an essential step in non-relativistic fermionic constructive quantum field theory to prove the absolute convergence of the perturbation expansion of correlation functions in terms of powers of the strength $u\\in \\mathbb{R}$ of the interparticle interaction. We provide, for large determinants of fermionic convariances, sharp bounds which hold for all (bounded and unbounded, the latter not being limited to semibounded) one-particle Hamiltonians. We find the smallest universal determinant bound to be exactly 1. In particular, the convergence of perturbation series at $u=0$ of any fermionic quantum field theory is ensured if the matrix entries, with respect to some fixed orthonormal basis, of the covariance and the interparticle interaction decay sufficiently fast. Our proofs use H\\"{o}lder inequalities for general non-commutative Lp-spaces derived by Araki and Masuda.

•Jan 1, 2017•11 citations

Correlation functions of one-dimensional strongly interacting two-component gases

Ovidiu I. Pâţu

We address the problem of calculating the correlation functions of one-dimensional two-component gases with strong repulsive contact interactions. The model considered in this paper describes particles with fractional statistics and in appropriate limits reduces to the Gaudin-Yang model or the spinor Bose gas. In the case of impenetrable particles we derive a Fredholm determinant representation for the temperature-, time-, and space-dependent correlation functions which is very easy to implement numerically and constitutes the starting point for the analytical investigation of the asymptotics. Making use of this determinant representation and the solution of an associated Riemann-Hilbert problem we derive the low-energy asymptotics of the correlators in the spin-incoherent regime characterized by near ground-state charge degrees of freedom but a highly thermally disordered spin sector. The asymptotics present features reminiscent of spin-charge separation with the spin part exponentially decaying in space separation and oscillating with a period proportional to the statistics parameter, while the charge part presents scaling with anomalous exponents which cannot be described by any unitary conformal field theory. The momentum distribution and the Fourier transform of the dynamical Green's function are asymmetrical for arbitrary statistics, a direct consequence of the broken space-reversal symmetry. Due to the exponential decay the momentum distribution $n(k)$ at zero temperature does not present algebraic singularities but the tails obey the universal decay ${lim}_{k\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}}n(k)\ensuremath{\sim}C/{k}^{4}$ with the amplitude $C$ given by Tan's contact. As a function of the statistics parameter the contact is a monotonic function reaching its minimum for the fermionic system and the maximum for the bosonic system.

arXiv:1910.10042•Physical Review A•Dec 23, 2019•22 citations
The asymptotics of an eigenfunction-correlation determinant for Dirac-δ\deltaδ perturbations
We prove the exact asymptotics of the scalar product of the ground states of two non-interacting Fermi gases confined to a 333-dimensional ball BLB_LBL​ of radius LLL in the thermodynamic limit, where the underlying one-particle operators differ by a Dirac-δ\deltaδ perturbation. More precisely, we show the algebraic decay of the correlation determinant ∣det⁡(⟨φjL,ψkL⟩)j,k=1,...,N∣2=L−ζ(E)+o(1)\big|\det\big(\langle\varphi_j^L, \psi_k^L\rangle\big)_{j,k=1,...,N}\big|^2= L^{-\zeta(E)+ \text{o}(1)}​det(⟨φjL​,ψkL​⟩)j,k=1,...,N​​2=L−ζ(E)+o(1), as N,L→∞N,L\to\inftyN,L→∞ and N/∣BL∣→ ρ>0N/|B_L|\to~ \rho>0N/∣BL​∣→ ρ>0, where φjL\varphi_j^LφjL​ and ψkL\psi_k^LψkL​ 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 ζ(E):=δ2(E)/π2\zeta(E):=\delta^2(\sqrt E)/{\pi^2}ζ(E):=δ2(E​)/π2. For an attractive Dirac-δ\deltaδ perturbation we conclude that the decay exponent 1π2∥arcsin⁡∣T(E)/2∣∥HS2\frac 1 {\pi^2}\Vert\arcsin |T(E)/2|\Vert^2_{\text{HS}}π21​∥arcsin∣T(E)/2∣∥HS2​ found in [GKMO14] does not provide a sharp upper bound on the decay of the correlation determinant.
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 BL of radius L in the thermodynamic limit, where the underlying one-particle operators differ by a Dirac-δ perturbation. More precisely, we show the algebraic decay of the correlation determinant det〈φjL,ψkL〉j,k=1,…,N2=L−ζ(E)+o(1), as N, L → ∞ and N/BL→ρ>0, where φjL and ψkL denote the lowest-energy eigenfunctions of the finite-volume one-particle Schrödinger operators. The decay exponent is given in terms of the s-wave scattering phase shift ζ(E)≔1π2δ2(E). For an attractive Dirac-δ perturbation, we conclude that the decay exponent 1π2arcsinT(E)/2HS2 found in the work of Gebert et al. [J. Spect. Theory (to be published); e-print arXiv:1407.2512] does not provide a sharp upper bound on the decay of the correlation determinant.
The goal of this thesis is a mathematical understanding of a phenomenon called Anderson's Orthogonality in the physics literature. Given two non-interacting fermionic systems which differ by an exterior short-range scattering potential, the scalar product of the corresponding ground states is predicted to decay algebraically in the thermodynamic limit. This decay is referred to as Anderson's orthogonality catastrophe in the physics literature and goes back to P.W.Anderson [Phys. Rev. Lett. 18:1049--1051] and is used to explain anomalies in the X-ray absorption spectrum of metals. We call this scalar product SNLS_N^LSNL​, where NNN refers to the particle number and LLL to the diameter of the considered fermionic system. This decay was proven in the works [Commun. Math. Phys. 329:979--998] and [arXiv:1407.2512] for rather general pairs of Schrödinger operators in arbitrary dimension d∈Nd\in\Nd∈N, i.e. ∣SNL∣2≤L−γ|S_N^L|^2\le L^{-\gamma}∣SNL​∣2≤L−γ in the thermodynamic limit N/Ld→ρ>0N/L^d\to \rho>0N/Ld→ρ>0 approaching a positive particle density. In the general case, the biggest found decay exponent is given by $\gamma=\frac 1 {\pi^2} \norm{\arcsin|T/2|}_{\text{HS}}$, where T refers to the scattering T-matrix. In this thesis, we prove such upper bounds in more general situations than considered in both [Commun. Math. Phys. 329:979--998] and [arXiv:1407.2512]. Furthermore, we provide the first rigorous proof of the exact asymptotics Anderson predicted. We prove that in the 333-dimensional toy model of a Dirac-δ\deltaδ perturbation that the exact decay exponent is given by ζ:=δ2/π2\zeta:= \delta^2/ \pi^2ζ:=δ2/π2. Here, δ\deltaδ refers to the s-wave scattering phase shift. In particular, this result shows that the previously found decay exponent γ\gammaγ does not provide the correct asymptotics of §NL\S_N^L§NL​ in general. Since the decay exponent is expressed in terms of scattering theory, these bounds depend on the existence of absolutely continuous spectrum of the underlying Schrödinger operators. We are able to deduce a different behavior in the contrary situation of Anderson localization. We prove the non-vanishing of the expectation value of the non-interacting many-body scalar product in the thermodynamic limit. Apart from the behavior of the scalar product of the non-interacting ground states, we also study the asymptotics of the difference of the ground-state energies. We show that this difference converges in the thermodynamic limit to the integral of the spectral-shift function up to the Fermi energy. Furthermore, we quantify the error for models on the half-axis and show that higher order error terms depend on the particular thermodynamic limit chosen.
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 -Fermion systems. We interpret the decay exponent \gamma in terms of scattering theory and find \gamma = \pi^{-2}\|\mathrm {arcsin}|T_E/2|\|_{\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.
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>0ν>0ν>0, confine fermions with total energy below ννν inside the interval [−L,L][-L,L][−L,L] and study the shift of the ground-state energy due to the potential VVV in the thermodynamic limit L→∞L\to\inftyL→∞. We show that the difference EL(ν)\mathcal{E}_L(ν)EL​(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/L1/L1/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-Δ+V−Δ+V on L2(R)L^2(\mathbb R)L2(R).
Efficiently bounding large determinants is an essential step in non-relativistic fermionic constructive quantum field theory to prove the absolute convergence of the perturbation expansion of correlation functions in terms of powers of the strength uinmathbbRu\\in \\mathbb{R}uinmathbbR of the interparticle interaction. We provide, for large determinants of fermionic convariances, sharp bounds which hold for all (bounded and unbounded, the latter not being limited to semibounded) one-particle Hamiltonians. We find the smallest universal determinant bound to be exactly 1. In particular, the convergence of perturbation series at u=0u=0u=0 of any fermionic quantum field theory is ensured if the matrix entries, with respect to some fixed orthonormal basis, of the covariance and the interparticle interaction decay sufficiently fast. Our proofs use H\\"{o}lder inequalities for general non-commutative Lp-spaces derived by Araki and Masuda.
We address the problem of calculating the correlation functions of one-dimensional two-component gases with strong repulsive contact interactions. The model considered in this paper describes particles with fractional statistics and in appropriate limits reduces to the Gaudin-Yang model or the spinor Bose gas. In the case of impenetrable particles we derive a Fredholm determinant representation for the temperature-, time-, and space-dependent correlation functions which is very easy to implement numerically and constitutes the starting point for the analytical investigation of the asymptotics. Making use of this determinant representation and the solution of an associated Riemann-Hilbert problem we derive the low-energy asymptotics of the correlators in the spin-incoherent regime characterized by near ground-state charge degrees of freedom but a highly thermally disordered spin sector. The asymptotics present features reminiscent of spin-charge separation with the spin part exponentially decaying in space separation and oscillating with a period proportional to the statistics parameter, while the charge part presents scaling with anomalous exponents which cannot be described by any unitary conformal field theory. The momentum distribution and the Fourier transform of the dynamical Green's function are asymmetrical for arbitrary statistics, a direct consequence of the broken space-reversal symmetry. Due to the exponential decay the momentum distribution n(k)n(k)n(k) at zero temperature does not present algebraic singularities but the tails obey the universal decay ${lim}_{k\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}}n(k)\ensuremath{\sim}C/{k}^{4}$ with the amplitude CCC given by Tan's contact. As a function of the statistics parameter the contact is a monotonic function reaching its minimum for the fermionic system and the maximum for the bosonic system.