## Lieb-Robinson Bounds for Multi-Commutators and Applications to Response TheoryLieb-Robinson bounds for multi-commutators are effective mathematical tools to handle analytic aspects of infinite volume dynamics of non-relativistic quantum particles with short-range, possibly time-dependent interactions.In particular, the existence of fundamental solutions is shown for those (non-autonomous) C*-dynamical systems for which the usual conditions found in standard theories of (parabolic or hyperbolic) non-autonomous evolution equations are not given. In mathematical physics, bounds on multi-commutators of an order higher than two can be used to study linear and non-linear responses of interacting particles to external perturbations. These bounds are derived for lattice fermions, in view of applications to microscopic quantum theory of electrical conduction discussed in this book. All results also apply to quantum spin systems, with obvious modifications. In order to make the results accessible to a wide audience, in particular to students in mathematics with little Physics background, basics of Quantum Mechanics are presented, keeping in mind its algebraic formulation. The C*-algebraic setting for lattice fermions, as well as the celebrated Lieb-Robinson bounds for commutators, are explained in detail, for completeness. |

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### Contents

1 | |

5 | |

4 LiebRobinson Bounds for Multicommutators | 31 |

5 LiebRobinson Bounds for Nonautonomous Dynamics | 63 |

6 Applications to Conductivity Measures | 89 |

103 | |

107 | |

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absolutely convergent algebraic formulation assertion Assume automorphisms Banach space bounded family bounded linear operators bounded operators bounds for multi-commutators BPH1 C∗-algebra CAR C∗–algebra Co-group compacta constant De Rt Corollary 5.2 defined dense set density elements existence Exponential decay fermion systems fermions finite dimensional finite-volume Fock space Gevrey Hamiltonian Heisenberg picture Hilbert space holds true increment infinite initial value problem interacting fermions irreducible representation k—automorphisms lattice Lemma Lieb–Robinson bounds non-autonomous evolution equation norm observables operators acting particle number perturbations Polynomial decay potential Quantum Mechanics quantum particles quantum spin quantum spin systems Response Theory satisfying Schrödinger self-adjoint short-range ſº so–called space H strongly continuous subset subspace symmetric derivation telescoping series Theorem 4.8 time-dependent Tºll uniformly unique vºn wave functions