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+34 946 567 842
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+34 946 567 842
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jbru@bcamath.org
Information of interest
- Orcid: 0000-0001-9159-9407
Other links
Personal website at IkerbasqueMy first researches have been related to mathematical studies of phase transitions associated with Bose gases. Since then, the bulk of my research has considerably widened and covers a scope from mathematical analyzes of the quantum many body problem to operator algebras, stochastic processes, differential equations, game theory, convex and functional analysis. I am particularly interested in the mathematical foundations of statistical mechanics with applications in the study of phase transitions and transport properties.
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C*-Algebraic Formulation of Quantum Mechanics
Bru, J.-B.
; de Siqueira Pedra, W. (2025)
This manuscript is based on a series of lectures given in the XX Jacques-Louis Lions Spanish-French School on Numerical Simulations in Physics & Engineering. While quantum mechanics is widely presented within the Hilbert ...
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Quadratic Hamiltonians in Fermionic Fock Spaces
Bru, J.-B.; Métraud, N.J.H. (2025)
Quadratic Hamiltonians are important in quantum field theory and quantum statistical mechanics. Their general studies, which go back to the sixties, are relatively incomplete for the fermionic case studied here. Following ...
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Non-Linear Operator-valued Elliptic Flows with Application to Quantum Field Theory
Bru, J.-B.; Métraud, N.J.H. (2025)
Differential equations on spaces of operators are very little developed in Mathematics, being in general very challenging. Here, we study a novel system of such (non-linear) differential equations. We show it has a unique ...
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Scattering and Pairing by Exchange Interactions
Bru, J.-B.; de Siqueira Pedra, W.; Ramer dos Santos, A. (2025)
Quantum interactions exchanging different types of particles play a pivotal rôle in quantum many-body theory, but they are not sufficiently investigated from a mathematical perspective. Here, we consider a system made of ...
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C*-Algebras and Mathematical Foundations of Quantum Statistical Mechanics
Bru, J.-B.; de Siqueira Pedra, W. (2023-06-19)
The present book grew from lecture notes we have written for participants of lectures on applications of C^{∗}-algebra theory to the foundations of quantum statistical mechanics, as well as a mini-course on thermodynamic ...
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A First Rigorous Attempt to Explain Charge Transport in a Protein-Ligand complex
Dempsey, R.M.; Uria-Albizuri, J.; Bru, J.-B.; Rodrigues, J. (2023-01-05)
Recent experimental evidence shows that when a protein (or peptide) binds to its ligand pair, the protein effectively “switches on” enabling long-range charge transport within the protein. Astonishingly, the protein-ligand ...
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From short-range to mean-field models in quantum lattices
de Siqueira Pedra, W.; Bru, J.-B.; Rodrigues, K. (2023-01-01)
Realistic effective interparticle interactions of quantum many-body systems are widely seen as being short-range. However, the rigorous mathematical analysis of this type of model turns out to be extremely difficult, in ...
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Thermodynamic Game and the Kac Limit in Quantum Lattices
Bru, J.-B.; de Siqueira Pedra, W.; Rodrigues, K. (2023-01-01)
A mathematically rigorous computation of the pressure and equilibrium states of important short-range quantum models on lattices (like the Hubbard model) to show possible phase transitions is generally elusive, beyond ...
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Classical dynamics from self-consistency equations in quantum mechanics
Bru, J.-B.; de Siqueira Pedra, W. (2022-05-09)
During the last three decades, P. Bóna has developed a non-linear generalization of quantum mechanics, based on symplectic structures for normal states and offering a general setting which is convenient to study the emergence ...
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Weak* Hypertopologies with Application to Genericity of Convex Sets
Bru, J.-B.; de Siqueira Pedra, W. (2022)
We propose a new class of hypertopologies, called here weak$^{\ast }$ hypertopologies, on the dual space $\mathcal{X}^{\ast }$ of a real or complex topological vector space $\mathcal{X}$. The most well-studied and well-known ...
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Quantum Dynamics Generated by Long-Range Interactions for Lattice Fermion and Quantum Spins
Bru, J.-B.; de Siqueira Pedra, W. (2021)
We study the macroscopic dynamics of fermion and quantum-spin systems with long-range, or mean-field, interactions, which turns out to be equivalent to an intricate combination of classical and short-range quantum dynamics. ...
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Large Deviations in Weakly Interacting Fermions - Generating Functions as Gaussian Berezin Integrals and Bounds on Large Pfaffians
Aza, N. J. B.; Bru, J.-B.; de Siqueira Pedra, W.; Müssnich, L. C. P. A. M. (2021)
We prove that the G\"{a}rtner--Ellis generating function of probability distributions associated with KMS states of weakly interacting fermions on the lattice can be written as the limit of logarithms of Gaussian Berezin ...
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Classical dynamics generated by long-range interactions for lattice fermions and quantum spins
Bru, J.-B.; de Siqueira Pedra, W. (2021)
We study the macroscopic dynamical properties of fermion and quantum-spin systems with long-range, or mean-field, interactions. The results obtained are far beyond previous ones and require the development of a mathematical ...
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Quantum Fluctuations and Large Deviation Principle for Microscopic Currents of Free Fermions in Disordered Media
Bru, J.-B.; de Siqueira Pedra, W.; Ratsimanetrimanana, A. (2020)
We contribute an extension of large-deviation results obtained in [N.J.B. Aza, J.-B. Bru, W. de Siqueira Pedra, A. Ratsimanetrimanana, J. Math. Pures Appl. 125 (2019) 209] on conductivity theory at atomic scale of free ...
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Macroscopic Dynamics of the Strong-Coupling BCS-Hubbard Model,
Bru, J.-B.; de Siqueira Pedra, W. (2020)
The aim of the current paper is to illustrate, in a simple example, our recent, very general, rigorous results on the dynamical properties of fermions and quantum-spin systems with long-range, or meanfield, interactions, ...
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Macroscopic Dynamics of the Strong-Coupling BCS-Hubbard Model
Bru, J.-B.; de Siqueira Pedra, W. (2020)
The aim of the current paper is to illustrate, in a simple example, our recent, very general, rigorous results on the dynamical properties of fermions and quantum-spin systems with long-range, or mean-field, interactions, ...
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Accuracy of Classical Conductivity Theory at Atomic Scales for Free Fermions in Disordered Media
Aza, N.J.B.; Bru, J.-B.; de Siqueira Pedra, W.; Ratsimanetrimanana, A. (2019-01-22)
The growing need for smaller electronic components has recently sparked the interest in the breakdown of the classical conductivity theory near the atomic scale, at which quantum effects should dominate. In 2012, experimental ...
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The Discreteness-driven Relaxation of Collisionless Gravitating Systems: Entropy Evolution in External Potentials, N-dependence, and the Role of Chaos
Beraldo e Silva, L.; de Siqueira Pedra, W.; Valluri, M.; Sodré, L.; Bru, J.-B. (2019-01-10)
We investigate the old problem of the fast relaxation of collisionless N-body systems that are collapsing or perturbed, emphasizing the importance of (noncollisional) discreteness effects. We integrate orbit ensembles in ...
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Isotropic Bipolaron-Fermion-Exchange Theory and Unconventional Pairing in Cuprate Superconductors
Bru, J.-B.; de Siqueira Pedra, W.; Delgado de Pasquale, A. (2018-12-10)
The discovery of high-temperature superconductors in 1986 represented a major experimental breakthrough (Nobel Prize 1987), but their theoretical explanation is still a subject of much debate. These materials have many ...
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Decay of Complex-time Determinantal and Pfaffian\ Correlation Functionals in Lattices
Aza, N.J.B.; Bru, J.-B.; de Siqueira Pedra, W. (2018-01-24)
We supplement the determinantal and Pfaffian bounds of Sims and Warzel (Commun Math Phys 347:903--931, 2016) for many-body localization of quasi-free fermions, by considering the high dimensional case and complex-time ...
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Universal bounds for large determinants from non-commutative Hölder inequalities in fermionic constructive quantum field theory
Bru, J.-B.; de Siqueira Pedra, W. (2017-08-02)
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 ...
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Isotropic Bipolaron-Fermion-Exchange Theory and Unconventional Pairing in Cuprate Superconductors
Bru, J.-B.; de Siqueira Pedra, W.; de Pasquale, A.D. (2017-05-03)
The discovery of high-temperature superconductors in 1986 represented a major experimental breakthrough (Nobel Prize 1987), but their theoretical explanation is still a subject of much debate. These materials have many ...
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Lieb–Robinson Bounds for Multi–Commutators and Applications to Response Theory
Bru, J.-B.; de Siqueira Pedra, W. (2017)
We generalize to multi-commutators the usual Lieb–Robinson bounds for commutators. In the spirit of constructive QFT, this is done so as to allow the use of combinatorics of minimally connected graphs (tree expansions) in ...
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Diagonalizing quadratic bosonic operators by non-autonomous flow equations volker bach
Bach, V.; Bru, J.-B. (2016-01-01)
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 ...
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Lieb–Robinson Bounds for Multi–Commutators and Applications to Response Theory
Bru, J.-B.; de Siqueira Pedra, W. (2016-01-01)
We generalize to multi–commutators the usual Lieb–Robinson bounds for commutators. In the spirit of constructive QFT, this is done so as to allow the use of combinatorics of minimally connected graphs (tree expan- sions) ...
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Microscopic Conductivity of Lattice Fermions at Equilibrium. Part II: Interacting Particles
Bru, J.-B.; de Siqueira Pedra, W. (2016-01-01)
We apply Lieb–Robinson bounds for multi-commutators we recently derived (Bru and de Siqueira Pedra, Lieb–Robinson bounds for multi-commutators and applications to response theory, 2015) to study the (possibly non-linear) ...
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Universal Bounds for Large Determinants from Non–Commutative Ho ̈lder Inequalities in Fermionic Constructive Quantum Field Theory
Bru, J.-B.; de Siqueira Pedra, W. (2016-01-01)
Efficiently bounding large determinants is an essential step in non–relati- vistic fermionic constructive quantum field theory, because, together with the summability of the interaction and the covariance, it implies the ...
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Microscopic conductivity of lattice fermions at equilibrium. I. Non-interacting particles
Bru, J.-B.; de Siqueira Pedra, W.; Hertling, C. (2015-12-31)
We consider free lattice fermions subjected to a static bounded potential and a timeand space-dependent electric field. For any bounded convex region R ⊂ ℠(d (d ≥ 1) of space, electric fields ε within R drive currents. ...
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AC-Conductivity Measure from Heat Production of Free Fermions in Disordered Media
Bru, J.-B.; de Siqueira Pedra, W.; Hertling, C. (2015-12-31)
We extend (Bru et al. in J Math Phys 56:051901-1-51, 2015) in order to study the linear response of free fermions on the lattice within a (independently and identically distributed) random potential to a macroscopic electric ...
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Characterization of the Quasi-Stationary State of an Impurity Driven by Monochromatic Light II: Microscopic Foundations
Bru, J.-B.; de Siqueira Pedra, W. (2015-12-31)
From quantum mechanical first principles only, we rigorously study the time-evolution of a N-level atom (impurity) interacting with an external monochromatic light source within an infinite system of free electrons at ...
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D-Wave pairing driven by bipolaric modes related to giant electron-phonon anomalies in high-Tc superconductors
Bru, J.-B.; de Pasquale, A.D.; de Siqueira Pedra, W. (2015-12-31)
Taking into account microscopic properties of most usual high-Tc superconductors, like cuprates, we define a class of microscopic model Hamiltonians for two fermions (electrons or holes) and one boson (bipolaron) on the ...
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Microscopic Conductivity of Lattice Fermions at Equilibrium. Part II: Interacting Particles
Bru, J.-B.; de Siqueira Pedra, W. (2015-12-31)
We apply Lieb–Robinson bounds for multi-commutators we recently derived (Bru and de Siqueira Pedra, Lieb–Robinson bounds for multi-commutators and applications to response theory, 2015) to study the (possibly non-linear) ...
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From the 2nd Law of Thermodynamics to AC–Conductivity Measures of Interacting Fermions in Disordered Media
Bru, J.-B.; de Siqueira Pedra, W. (2015-05-20)
We study the dynamics of interacting lattice fermions with random hopping amplitudes and random static potentials, in presence of time-dependent electromagnetic fields. The interparticle interaction is short-range and ...
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Macroscopic conductivity of free fermions in disordered media
Bru, J.-B.; de Siqueira Pedra, W.; Hertling, C. (2014-12-31)
We conclude our analysis of the linear response of charge transport in lattice systems of free fermions subjected to a random potential by deriving general mathematical properties of its conductivity at the macroscopic ...
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Heat production of noninteracting fermions subjected to electric fields
Bru, J.-B.; de Siqueira Pedra, W.; Hertling, C. (2014-07-21)
Electric resistance in conducting media is related to heat (or entropy) production in the presence of electric fields. In this paper, by using Araki's relative entropy for states, we mathematically define and analyze the ...
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Non-cooperative Equilibria of Fermi Systems With Long Range Interactions
Bru, J.-B.; de Siqueira Pedra, W. (2013-07)
We define a Banach space $\mathcal{M}_{1}$ of models for fermions or quantum spins in the lattice with long range interactions and explicit the structure of (generalized) equilibrium states for any $\mathfrak{m}\in ...