arXiv subjects

Volker Bach

Publications and source records attributed to Volker Bach.

At least 19 recordsLinked to original sources

arXiv2026

Bounds on the Bogoliubov--Hartree--Fock Energy of the Pauli--Fierz Hamiltonian

A variational analysis of the Bogoliubov--Hartree--Fock (BHF) energy of the translation-invariant, spinless Pauli--Fierz Hamiltonian with massless dispersion relation built up on work of the first author, Breteaux, and Tzaneteas (2013) and of the first author and Hach (2022) is presented. The main results are lower and upper bounds on the BHF energy for fixed total momentum expressed through simpler variational problems defined on the space of positive Hilbert--Schmidt operators and a new variational formulation of the upper bound for zero total momentum. Specifically, we introduce a change of variables which considerably simplifies the energy functional and the derivation of its stationarity condition.

math-ph↗

arXiv2026

On the Ultraviolet Problem for the Ground State Energy of the Translation-Invariant Pauli--Fierz Model at Zero Total Momentum

We study the ground state energy of the Pauli--Fierz model in the absence of external potentials. We consider the fiber decomposition of the Pauli--Fierz operator with respect to the spectral values, $p$, of the total momentum operator and focus on the case $p = 0$. The corresponding variational problem is analyzed to estimate the dependence of the ground state energy on the ultraviolet cutoff $\Lambda$. We employ a Bogoliubov--Hartree--Fock approximation using pure, quasifree states generated by Bogolubov transformations (parametrized by a positive Hilbert--Schmidt operator $z$) and Weyl transformations (parametrized by a vector $\eta$) applied to the vacuum. We prove that the resulting energy functional is not a convex function of $\eta$ and $z$. We identify the non-convex term and remove it from the energy functional. The modified functional retains the full interaction term and is shown to be strictly convex. We study the ground state of the modified functional and prove the existence of a unique minimizer. Furthermore, we construct an explicit partial minimizer (with respect to $\eta$, for fixed $z$), which allows us to eliminate $z$ and reduce the minimization problem to a single variable, $\eta$. Finally, we estimate the minimum of the modified energy functional in terms of the ultraviolet cutoff $\Lambda$ and demonstrate that, up to a constant factor, it grows asymptotically as $\Lambda^{3/2}$, as $\Lambda \to \infty$.

math-ph↗

arXiv2025

The Spectral Renormalization Flow Based on the Smooth Feshbach--Schur Map: The Introduction of the Semi-Group Property

The spectral renormalization method is a powerful mathematical tool that is prominently used in spectral theory in the context of low-energy quantum field theory and its original introduction in [5, 6] constituted a milestone in the field. Inspired by physics, this method is usually called renormalization group, even though it is not a group nor a semigroup (or, more properly, a flow). It was only in 2015 in [1] when a flow (or semigroup) structure was first introduced using an innovative definition of the renormalization of spectral parameters. The spectral renormalization flow in [1], however, is not compatible with the smooth Feshbach--Schur map (this is stated as an open problem in [1]), which is a lamentable weakness because its smoothness is a key feature that significantly simplifies the proofs and makes it the preferred tool in most of the literature. In this paper we solve this open problem introducing a spectral renormalization flow based on the smooth Feshbach--Schur map.

math-ph↗

arXiv2022

On Relative Bounds for Interacting Fermion Operators

We consider a Hubbard model with nearest neighbor interaction on a discrete $d$-dimensional torus of length $L$ around its Hartree-Fock ground state and derive relative bounds of the effective interaction with respect to the effective kinetic energy. It is shown that there are no relative bounds uniform in $L$.

math-ph↗

arXiv2022

Hartree--Fock Theory, Lieb's Variational Principle, and their Generalizations

Hartree--Fock theory in quantum mechanics is reviewed, from the proposal of the Hartree--Fock approximation right after quantum mechanics was formulated to its applications in modern physics. This includes the description of traditional Hartree--Fock theory in quantum chemistry, its generalizations of various kinds, and its importance for predicting the presence of symmetry breaking, or the absence thereof.

quant-ph↗

arXiv2020

Threshold effects of the two-particle Schrödinger operators on lattices

We consider a wide class of the two-particle Schrödinger operators $H_μ(k)=H_{0}(k)+μV, \,μ>0,$ with a fixed two-particle quasi-momentum $k$ in the $d$ -dimensional torus $\mathbb{T}^d$, associated to the Bose-Hubbard hamiltonian $H_μ$ of a system of two identical quantum-mechanical particles (bosons) on the $d$- dimensional hypercubic lattice $\mathbb{Z}% ^d$ interacting via short-range pair potentials. We study the existence of eigenvalues of $H_μ(k)$ below the threshold of the essential spectrum depending on the interaction energy $μ>0$ and the quasi-momentum $k\in \mathbb{T}^d$ of particles. We prove that the threshold (bottom of the essential spectrum), as a singular point (a threshold resonance or a threshold eigenvalue), creates eigenvalues below the essential spectrum under perturbations of both the coupling constant $μ>0$ and the quasi-momentum $k$ of the particles. Moreover, we show that if the threshold is a regular point, then it does not create any eigenvalues under small perturbations of the coupling constant $μ>0$ and the quasi-momentum $k$.

math.SP↗

arXiv2020

On the Ultraviolet Limit of the Pauli-Fierz Hamiltonian in the Lieb-Loss Model

Two decades ago, Lieb and Loss proposed to approximate the ground state energy of a free, nonrelativistic electron coupled to the quantized radiation field by the infimum $E_{α, Λ}$ of all expectation values $\langle ϕ_{el} \otimes ψ_{ph} | H_{α, Λ} (ϕ_{el} \otimes ψ_{ph}) \rangle$, where $H_{α, Λ}$ is the corresponding Hamiltonian with fine structure constant $α>0$ and ultraviolet cutoff $Λ< \infty$, and $ϕ_{el}$ and $ψ_{ph}$ are normalized electron and photon wave functions, respectively. Lieb and Loss showed that $c α^{1/2} Λ^{3/2} \leq E_{α, Λ} \leq c^{-1} α^{2/7} Λ^{12/7}$ for some constant $c >0$. In the present paper we prove the existence of a constant $C < \infty$, such that \begin{align*} \bigg| \frac{E_{α, Λ}}{F[1] \, α^{2/7} \, Λ^{12/7}} - 1 \bigg| \ \leq \ C \, α^{4/105} \, Λ^{-4/105} \end{align*} holds true, where $F[1] >0$ is an explicit universal number. This result shows that Lieb and Loss' upper bound is actually sharp and gives the asymptotics of $E_{α, Λ}$ uniformly in the limit $α\to 0$ and in the ultraviolet limit $Λ\to \infty$.

math-ph↗

arXiv2019

Orthogonalization of fermion k-body operators and representability

The reduced k-particle density matrix of a density matrix on finite-dimensional, fermion Fock space can be defined as the image under the orthogonal projection in the Hilbert-Schmidt geometry onto the space of k-body observables. A proper understanding of this projection is therefore intimately related to the representability problem, a long-standing open problem in computational quantum chemistry. Given an orthonormal basis in the finite-dimensional one-particle Hilbert space, we explicitly construct an orthonormal basis of the space of Fock space operators which restricts to an orthonormal basis of the space of k-body operators for all k.

math-ph↗

arXiv2018

The time-dependent Hartree-Fock-Bogoliubov equations for Bosons

In this article, we use quasifree reduction to derive the time-dependent Hartree-Fock-Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose-Einstein condensate in $\mathbb R^d$. We prove global well-posedness for the HFB equations for sufficiently regular pair interaction potentials, and establish key conservation laws. Moreover, we show that the solutions to the HFB equations exhibit a symplectic structure, and have a form reminiscent of a Hamiltonian system. In particular, this is used to relate the HFB equations to the HFB eigenvalue equations encountered in the physics literature. Furthermore, we construct the Gibbs states at positive temperature associated with the HFB equations, and establish criteria for the emergence of Bose-Einstein condensation.

math-ph↗

arXiv2018

On the Hartree-Fock-Bogoliubov equations

We review some results of our paper arXiv:1602.05171v2 on the "nonlinear quasifree approximation" to the many-body Schrödinger dynamics of Bose gases. In that paper, we derive, with the help of this approximation, the time-dependent Hartree-Fock-Bogoliubov (HFB) equations, providing an approximate description of the dynamics of quantum fluctuations around a Bose-Einstein condensate and study properties of these equations.

math-ph↗

arXiv2017

Bounds on the Pure Point Spectrum of Lattice Schrödinger Operators

In dimension $d\geq 3$, a variational principle for the size of the pure point spectrum of (discrete) Schrödinger operators $H(\mathfrak{e},V)$ on the hypercubic lattice $\mathbb{Z}^{d}$, with dispersion relation $\mathfrak{e}$ and potential $V$, is established. The dispersion relation $\mathfrak{e}$ is assumed to be a Morse function and the potential $V(x)$ to decay faster than $|x|^{-2(d+3)}$, but not necessarily to be of definite sign. Our estimate on the size of the pure-point spectrum yields the absence of embedded and threshold eigenvalues of $H(\mathfrak{e},V)$ for a class ot potentials of this kind. The proof of the variational principle is based on a limiting absorption principle combined with a positive commutator (Mourre) estimate, and a Virial theorem. A further observation of crucial importance for our argument is that, for any selfadjoint operator $B$ and positive number $λ>0$, the number of negative eigenvalues of $λB$ is independent of $λ$.

math-ph↗

arXiv2017

Bounds on the Discrete Spectrum of Lattice Schrödinger Operators

We discuss the validity of the Weyl asymptotics -- in the sense of two-sided bounds -- for the size of the discrete spectrum of (discrete) Schrödinger operators on the $d$--dimensional, $d\geq 1$, cubic lattice $\mathbb{Z}^{d}$ at large couplings. We show that the Weyl asymptotics can be violated in any spatial dimension $d\geq 1$ -- even if the semi-classical number of bound states is finite. Furthermore, we prove for all dimensions $d\geq 1$ that, for potentials well-behaved at infinity and fulfilling suitable decay conditions, the Weyl asymptotics always hold. These decay conditions are mild in the case $d\geq 3$, while stronger for $d=1,2$. It is well-known that the semi-classical number of bound states is -- up to a constant -- always an upper bound on the size of the discrete spectrum of Schrödinger operators if $d\geq 3$. We show here how to construct general upper bounds on the number of bound states of Schrödinger operators on $\mathbb{Z}^{d}$ from semi-classical quantities in all space dimensions $d\geq 1$ and independently of the positivity-improving property of the free Hamiltonian.

math-ph↗

arXiv2017

Suppression of Decoherence of a Spin-Boson System by Time-Periodic Control

We consider a finite-dimensional quantum system coupled to the bosonic radiation field and subject to a time-periodic control operator. Assuming the validity of a certain dynamic decoupling condition we approximate the system's time evolution with respect to the non-interacting dynamics. For sufficiently small coupling constants $g$ and control periods $T$ we show that a certain deviation of coupled and uncoupled propagator may be estimated by $\mathcal{O}(gt \, T)$. Our approach relies on the concept of Kato stability and general theory on non-autonomous linear evolution equations.

math-ph↗

arXiv2016

Diagonalizing Quadratic Bosonic Operators by Non-Autonomous Flow Equation

We study a non-autonomous, non-linear evolution equation on the space of operators on a complex Hilbert space. We specify assumptions that ensure the global existence of its solutions and allow us to derive its asymptotics at temporal infinity. We demonstrate that these assumptions are optimal in a suitable sense and more general than those used before. The evolution equation derives from the Brocket-Wegner flow that was proposed to diagonalize matrices and operators by a strongly continuous unitary flow. In fact, the solution of the non-linear flow equation leads to a diagonalization of Hamiltonian operators in boson quantum field theory which are quadratic in the field.

math-ph↗

arXiv2016

Existence of Ground State Eigenvalues for the Spin-Boson Model with Critical Infrared Divergence and Multiscale Analysis

A two-level atom coupled to the radiation field is studied. First principles in physics suggest that the coupling function, representing the interaction between the atom and the radiation field, behaves like $\vert k \vert^{- 1/2}$, as the photon momentum k tends to zero. Previous results on non-existence of ground state eigenvalues suggest that in the most general case binding does not occur in the spin-boson model, i.e., the minimal energy of the atom-photon system is not an eigenvalue of the energy operator. Hasler and Herbst have shown [12], however, that under the additional hypothesis that the coupling function be off-diagonal -which is customary to assume-binding does indeed occur. In this paper an alternative proof of binding in case of off-diagonal coupling is given, i.e., it is proven that, if the coupling function is off-diagonal, the ground state energy of the spin-boson model is an eigenvalue of the Hamiltonian. We develop a multiscale method that can be applied in the situation we study, identifying a new key symmetry operator which we use to demonstrate that the most singular terms appearing in the multiscale analysis vanish.

math-ph↗

arXiv2015

Kinetic Energy Estimates for the Accuracy of the Time-Dependent Hartree-Fock Approximation with Coulomb Interaction

We study the time evolution of a system of $N$ spinless fermions in $\mathbb{R}^3$ which interact through a pair potential, e.g., the Coulomb potential. We compare the dynamics given by the solution to Schr{ö}dinger's equation with the time-dependent Hartree-Fock approximation, and we give an estimate for the accuracy of this approximation in terms of the kinetic energy of the system. This leads, in turn, to bounds in terms of the initial total energy of the system.

math-ph↗

arXiv2014

On Some Open Problems in Many-Electron Theory

Mel Levy and Elliott Lieb are two of the most prominent researchers who have dedicated their efforts to the investigation of fundamental questions in many-electron theory. Their results have not only revolutionized the theoretical approach of the field, but, directly or indirectly, allowed for a quantum jump in the computational treatment of realistic systems as well. For this reason, at the conclusion of our book where the subject is treated across different disciplines, we have asked Mel Levy and Elliott Lieb to provide us with some open problems, which they believe will be a worth challenge for the future also in the perspective of a synergy among the various disciplines.

quant-ph↗

arXiv2014

Suppression of Decoherence by Periodic Forcing

We consider a finite-dimensional quantum system coupled to a thermal reservoir and subject to a time-periodic, energy conserving forcing. We show that, if a certain dynamical decoupling condition is fulfilled, then the periodic forcing counteracts the decoherence induced by the reservoir: for small system-reservoir coupling $λ$ and small forcing period $T$, the system dynamics is approximated by an energy conserving and non-dissipative dynamics, which preserves coherences. For times up to order $(λT)^{-1}$, the difference between the true and approximated dynamics is of size $λ+T$. Our approach is rigorous and combines Floquet and spectral deformation theory. We illustrate our results on the spin-fermion model and recover previously known, heuristically obtained results.

quant-ph↗