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We study the number $N_{<0}(H_s)$ of negative eigenvalues, counting multiplicities, of the fractional Schr\"odinger operator $H_s=(-\Delta)^s-V(x)$ on $L^2(\mathbb{R}^d)$, for any $d\ge1$ and $s\ge d/2$. We prove a bound on $N_{<0}(H_s)$ which depends on $s-d/2$ being either an integer or not, the critical case $s=d/2$ requiring a further analysis. Our proof relies on a splitting of the Birman-Schwinger operator associated to this spectral problem into low- and high-energies parts, a projection of the low-energies part onto a suitable subspace, and, in the critical case $s=d/2$, a Cwikel-type estimate in the weak trace ideal $\mathcal{L}^{2,\infty}$ to handle the high-energies part.

We consider Markovian open quantum dynamics (MOQD). We show that, up to small-probability tails, the supports of quantum states evolving under such dynamics propagate with finite speed in any finite-energy subspace. More precisely, we prove that if the initial quantum state is localized in space, then any finite-energy part of the solution of the von Neumann-Lindblad equation is approximately localized inside an energy-dependent light cone. We also obtain an explicit upper bound for the slope of this light cone., Comment: 43 pages

We prove maximal speed estimates for nonlinear quantum propagation in the context of the Hartree equation. More precisely, under some regularity and integrability assumptions on the pair (convolution) potential, we construct a set of energy and space localized initial conditions such that, up to time-decaying tails, solutions starting in this set stay within the light cone of the corresponding initial datum. We quantify precisely the light cone speed, and hence the speed of nonlinear propagation, in terms of the momentum of the initial state., Comment: 38 pages. Appendices D and E are adapted from arXiv:2011.04570. To appear in Communications in Partial Differential Equations

We consider a non-relativistic electron bound by an external potential and coupled to the quantized electromagnetic field in the standard model of non-relativistic QED. We compute the energy functional of product states of the form $u\otimes \Psi_f$, where $u$ is a normalized state for the electron and $\Psi_f$ is a coherent state in Fock space for the photon field. The minimization of this functional yields a Maxwell--Schr{\"o}dinger system up to a trivial renormalization. We prove the existence of a ground state under general conditions on the external potential and the coupling. In particular, neither an ultraviolet cutoff nor an infrared cutoff needs to be imposed. Our results provide the convergence in the ultraviolet limit and the second-order asymptotic expansion in the coupling constant of the ground state energy of Maxwell--Schr\"odinger systems.

We prove a maximal velocity bound for the dynamics of Markovian open quantum systems. The dynamics are described by one-parameter semi-groups of quantum channels satisfying the von Neumann-Lindblad equation. Our result says that dynamically evolving states are contained inside a suitable light cone up to polynomial errors. We also give a bound on the slope of the light cone, i.e., the maximal propagation speed. The result implies an upper bound on the speed of propagation of local perturbations of stationary states in open quantum systems.

We consider a spinless, non-relativistic particle bound by an external potential and linearly coupled to a quantized radiation field. The energy $\mathcal{E}(u,f)$ of product states of the form $u\otimes \Psi_f$, where $u$ is a normalized state for the particle and $\Psi_f$ is a coherent state in Fock space for the field, gives the energy of a Klein-Gordon--Schr\''odinger system. We minimize the functional $\mathcal{E}(u,f)$ on its natural energy space. We prove the existence and uniqueness of a ground state under general conditions on the coupling function. In particular, neither an ultraviolet cutoff nor an infrared cutoff is imposed. Our results establish the convergence in the ultraviolet limit of both the ground state and ground state energy of the Klein-Gordon--Schr\''odinger energy functional, and provide the second-order asymptotic expansion of the ground state energy at small coupling.

We consider non-self-adjoint operators in Hilbert spaces of the form $H=H_0+CWC$, where $H_0$ is self-adjoint, $W$ is bounded and $C$ is a metric operator, $C$ bounded and relatively compact with respect to $H_0$. We suppose that $C(H_0-z)^{-1}C$ is uniformly bounded in $z\in\mathbb{C}\setminus\mathbb{R}$. We define the spectral singularities of $H$ as the points of the essential spectrum $\lambda\in\sigma_{\mathrm{ess}}(H)$ such that $C(H\pm i\varepsilon)^{-1}CW$ does not have a limit as $\varepsilon\to0^+$. We prove that the spectral singularities of $H$ are in one-to-one correspondence with the eigenvalues, associated to resonant states, of an extension of $H$ to a larger Hilbert space. Next, we show that the asymptotically disappearing states for $H$, i.e. the set of vectors $\varphi$ such that $e^{\pm itH}\varphi\to0$ as $t\to\infty$, coincide with the generalized eigenstates of $H$ corresponding to eigenvalues $\lambda\in\mathbb{C}$, $\mp\mathrm{Im}(\lambda)>0$. Finally, we define the absolutely continuous spectral subspace of $H$ and show that it satisfies $\mathcal{H}_{\mathrm{ac}}(H)=\mathcal{H}_{\mathrm{p}}(H^*)^\perp$, where $\mathcal{H}_{\mathrm{p}}(H^*)$ stands for the point spectrum of $H^*$. We thus obtain a direct sum decomposition of the Hilbert spaces in terms of spectral subspaces of $H$. One of the main ingredients of our proofs is a spectral resolution formula for a bounded operator $r(H)$ regularizing the identity at spectral singularities. Our results apply to Schr\"odinger operators with complex potentials., Comment: 41 pages, 4 figures

We study perturbed Bessel operators $L_{m^2}=- \partial^2_x + ( m^2 - \frac14 )\frac{1}{x^2} + Q(x)$ on $L^2]0,\infty[$, where $m\in\mathbb{C}$ and $Q$ is a complex locally integrable potential. Assuming that $Q$ is integrable near $\infty$ and $x\mapsto x^{1-\varepsilon}Q(x)$ is integrable near $0$, with $\varepsilon\ge0$, we construct solutions to $L_{m^2} f = - k^2 f$ with prescribed behaviors near $0$. The special cases $m=0$ and $k=0$ are included in our analysis. Our proof relies on mapping properties of various Green's operators of the unperturbed Bessel operator. Then we determine all closed realizations of $L_{m^2}$ and show that they can be organized as holomorphic families of closed operators., Comment: 67 pages

The Lieb-Robinson bound asserts the existence of a maximal propagation speed for the quantum dynamics of lattice spin systems. Such general bounds are not available for most bosonic lattice gases due to their unbounded local interactions. Here we establish for the first time a general ballistic upper bound on macroscopic particle transport in the paradigmatic Bose-Hubbard model. The bound is the first to cover a broad class of initial states with positive density including Mott states, which resolves a longstanding open problem. It applies to Bose-Hubbard type models on any lattice with not too long-ranged hopping. The proof is rigorous and rests on controlling the time evolution of a new kind of adiabatic spacetime localization observable via iterative differential inequalities., Comment: 18 pages, 3 figures

We consider the dynamics of the Bose-Hubbard model on general lattices and prove a Lieb-Robinson bound for observables whose supports are separated by an initially almost particle-free region. We further obtain a maximal velocity bound for particle transport through an initially empty region which also applies to long-range hopping. Our techniques originate in the proofs of maximal velocity bounds for Schr\"odinger operators and scattering theory in non-relativistic QED., Comment: 29 pages, to appear in CMP

We review recent results obtained in the scattering theory of dissipative quantum systems representing the long-time evolution of a system $S$ interacting with another system $S'$ and susceptible of being absorbed by $S'$. The effective dynamics of $S$ is generated by an operator of the form $H = H_0 + V - \mathrm{i} C^* C$ on the Hilbert space of the pure states of $S$, where $H_0$ is the self-adjoint generator of the free dynamics of $S$, $V$ is symmetric and $C$ is bounded. The main example is a neutron interacting with a nucleus in the nuclear optical model. We recall the basic objects of the scattering theory for the pair $(H,H_0)$, as well as the results, proven in arXiv:1703.09018 and arXiv:1808.09179, on the spectral singularities of $H$ and the asymptotic completeness of the wave operators. Next, for the nuclear optical model, we show that asymptotic completeness generically holds., Comment: Contribution to the proceedings of QMATH 14: Mathematical Results in Quantum Physics. Contains an overview of arXiv:1703.09018, arXiv:1808.09179 and the proof of a new result. 18 pages, 1 figure

We consider hamiltonian models representing an arbitrary number of spin $1/2$ fermion quantum fields interacting through arbitrary processes of creation or annihilation of particles. The fields may be massive or massless. The interaction form factors are supposed to satisfy some regularity conditions in both position and momentum space. Without any restriction on the strength of the interaction, we prove that the Hamiltonian identifies to a self-adjoint operator on a tensor product of anti-symmetric Fock spaces and we establish the existence of a ground state. Our results rely on new interpolated $N_\tau$ estimates. They apply to models arising from the Fermi theory of weak interactions, with ultraviolet and spatial cut-offs., Comment: 26 pages

The Spin-Boson model describes a two-level quantum system that interacts with a second-quantized boson scalar field. Recently the relation between the integral kernel of the scattering matrix and the resonance in this model has been established in [14] for the case of massless bosons. In the present work, we treat the massive case. On the one hand, one might rightfully expect that the massive case is easier to handle since, in contrast to the massless case, the corresponding Hamiltonian features a spectral gap. On the other hand, it turns out that the non-zero boson mass introduces a new complication as the spectrum of the complex dilated, free Hamiltonian exhibits lines of spectrum attached to every multiple of the boson rest mass energy starting from the ground and excited state energies. This leads to an absence of decay of the corresponding complex dilated resolvent close to the real line, which, in [14], was a crucial ingredient to control the time evolution in the scattering regime. With the new strategy presented here, we provide a proof of an analogous formula for the scattering kernel as compared to the massless case and use the opportunity to provide the required spectral information by a Mourre theory argument combined with a suitable application of the Feshbach-Schur map instead of complex dilation., Comment: 49 pages

We consider mathematical models of the weak decay of the vector bosons $W^{\pm}$ into leptons. The free quantum field hamiltonian is perturbed by an interaction term from the standard model of particle physics. After the introduction of high energy and spatial cut-offs, the total quantum hamiltonian defines a self-adjoint operator on a tensor product of Fock spaces. We study the scattering theory for such models. First, the masses of the neutrinos are supposed to be positive: for all values of the coupling constant, we prove asymptotic completeness of the wave operators. In a second model, neutrinos are treated as massless particles and we consider a simpler interaction Hamiltonian: for small enough values of the coupling constant, we prove again asymptotic completeness, using singular Mourre's theory, suitable propagation estimates and the conservation of the difference of some number operators.

We consider a quantum system S interacting with another system S and susceptible of being absorbed by S. The effective, dissipative dynamics of S is supposed to be generated by an abstract pseudo-Hamiltonian of the form H = H0 + V -- iC * C. The generator of the free dynamics, H0, is self-adjoint, V is symmetric and C is bounded. We study the scattering theory for the pair of operators (H, H0). We establish a representation formula for the scattering matrices and identify a necessary and sufficient condition to their invertibility. This condition rests on a suitable notion of spectral singularity. Our main application is the nuclear optical model, where H is a dissipative Schr{\"o}dinger operator and spectral singularities correspond to real resonances.

We consider an abstract pseudo-Hamiltonian for the nuclear optical model, given by a dissipative operator of the form $H = H_V - i C^* C$, where $H_V = H_0 + V$ is self-adjoint and $C$ is a bounded operator. We study the wave operators associated to $H$ and $H_0$. We prove that they are asymptotically complete if and only if $H$ does not have spectral singularities on the real axis. For Schr\"odinger operators, the spectral singularities correspond to real resonances., Comment: 48 pages

We consider a mathematical model for the weak decay of the intermediate boson $Z^0$ into neutrinos and antineutrinos. We prove that the total Hamiltonian has a unique ground state in Fock space and we establish a limiting absorption principle, local decay and a property of relaxation to the ground state for initial states and observables suitably localized in energy and position. Our proofs rest, in particular, on Mourre's theory and a low-energy decomposition., Comment: 42 pages

We study scattering theory for a quantum-mechanical system consisting of a particle scattered off a dynamical target that occupies a compact region in position space. After taking a trace over the degrees of freedom of the target, the dynamics of the particle is generated by a Lindbladian acting on the space of trace-class operators. We study scattering theory for a general class of Lindbladians with bounded interaction terms. First, we consider models where a particle approaching the target is always re-emitted by the target. Then we study models where the particle may be captured by the target. An important ingredient of our analysis is a scattering theory for dissipative operators on Hilbert space., Comment: 32 pages

An otherwise free classical particle moving through an extended spatially homogeneous medium with which it may exchange energy and momentum will undergo a frictional drag force in the direction opposite to its velocity with a magnitude which is typically proportional to a power of its speed. We study here the quantum equivalent of a classical Hamiltonian model for this friction phenomenon that was proposed in [11]. More precisely, we study the spectral properties of the quantum Hamiltonian and compare the quantum and classical situations. Under suitable conditions on the infrared behaviour of the model, we prove that the Hamiltonian at fixed total momentum has no ground state except when the total momentum vanishes, and that its spectrum is otherwise absolutely continuous., Comment: 40 pages

The notion of "closed systems" in Quantum Mechanics is discussed. For this purpose, we study two models of a quantum-mechanical system $P$ spatially far separated from the "rest of the universe" $Q$. Under reasonable assumptions on the interaction between $P$ and $Q$, we show that the system $P$ behaves as a closed system if the initial state of $P \vee Q$ belongs to a large class of states, including ones exhibiting entanglement between $P$ and $Q$. We use our results to illustrate the non-deterministic nature of quantum mechanics. Studying a specific example, we show that assigning an initial state and a unitary time evolution to a quantum system is generally not sufficient to predict the results of a measurement with certainty., Comment: Typos corrected. Accepted for publication in Ann. Henri Poincar\'e

A simple model of an atom interacting with the quantized electromagnetic field is studied. The atom has a finite mass $m$, finitely many excited states and an electric dipole moment, $\vec{d}_0 = -\lambda_{0} \vec{d}$, where $\| d^{i}\| = 1,$ $ i=1,2,3,$ and $\lambda_0$ is proportional to the elementary electric charge. The interaction of the atom with the radiation field is described with the help of the Ritz Hamiltonian, $-\vec{d}_0\cdot \vec{E}$, where $\vec{E}$ is the electric field, cut off at large frequencies. A mathematical study of the Lamb shift, the decay channels and the life times of the excited states of the atom is presented. It is rigorously proven that these quantities are analytic functions of the momentum $\vec{p}$ of the atom and of the coupling constant $\lambda_0$, provided $|\vec{p}| < mc$ and $| \Im\vec{p} |$ and $| \lambda_{0} |$ are sufficiently small. The proof relies on a somewhat novel inductive construction involving a sequence of `smooth Feshbach-Schur maps' applied to a complex dilatation of the original Hamiltonian, which yields an algorithm for the calculation of resonance energies that converges super-exponentially fast., Comment: Small typos and inconsistencies corrected. Accepted for publication in CMP

We consider a Hamiltonian describing the weak decay of the massive vector boson Z0 into electrons and positrons. We show that the spectrum of the Hamiltonian is composed of a unique isolated ground state and a semi-axis of essential spectrum. Using a suitable extension of Mourre's theory, we prove that the essential spectrum below the boson mass is purely absolutely continuous., Comment: 45 pages

We study a neutral atom with a non-vanishing electric dipole moment coupled to the quantized electromagnetic field. For a sufficiently small dipole moment and small momentum, the one-particle (self-) energy of an atom is proven to be a real-analytic function of its momentum. The main ingredient of our proof is a suitable form of the Feshbach-Schur spectral renormalization group., Comment: Small typos and inconsistencies corrected. Accepted for publication in J. Funct. Anal

In this note we provide details of the proofs of the main results of our paper [19] to the standard model of non-relativistic quantum electrodynamics in which particles are minimally coupled to the quantized electromagnetic field at energies below the ionization threshold. Recall that in [19] we proved several lower bounds on the growth of the distance of the escaping photons/phonons to the particle system. Using some of these results, we proved asymptotic completeness (for Rayleigh scattering) on the states for which the expectation of the photon/phonon number is bounded uniformly in time. However, we provided details only for the phonon case., Comment: 12 pages. arXiv admin note: text overlap with arXiv:1202.6151

We discuss photon number bounds for a system of non-relativistic particles coupled to the quantized electromagnetic field (non-relativistic QED), below the ionization threshold. Such a bound was assumed in the proof of asymptotic completeness for Rayleigh scattering in our paper [3] (Condition (1.20) of Theorem 1.3 in [3]). We show how this assumption can be weakened and verified for a class of hamiltonians., Comment: 6 pages

We prove several minimal photon velocity estimates below the ionization threshold for a particle system coupled to the quantized electromagnetic or phonon field. Using some of these results, we prove the asymptotic completeness (for the Rayleigh scattering) on the states for which the expectation of the photon number is uniformly bounded., Comment: 35 pages

In the framework of non-relativistic QED, we show that the renormalized mass of the electron (after having taken into account radiative corrections) appears as the kinematic mass in its response to an external potential force. Specifically, we study the dynamics of an electron in a slowly varying external potential and with slowly varying initial conditions and prove that, for a long time, it is accurately described by an associated effective dynamics of a Schr\"odinger electron in the same external potential and for the same initial data, with a kinetic energy operator determined by the renormalized dispersion law of the translation-invariant QED model., Comment: 22 pages, AMS Latex

We consider a free hydrogen atom composed of a spin-1/2 nucleus and a spin-1/2 electron in the standard model of non-relativistic QED. We study the Pauli-Fierz Hamiltonian associated with this system at a fixed total momentum. For small enough values of the fine-structure constant, we prove that the ground state is unique. This result reflects the hyperfine structure of the hydrogen atom ground state., Comment: 22 pages, 3 figures

We consider the problem of propagation of photons in the quantum theory of non-relativistic matter coupled to electromagnetic radiation, which is, presently, the only consistent quantum theory of matter and radiation. Assuming that the matter system is in a localized state (i.e for energies below the ionization threshold), we show that the probability to find photons at time t at the distance greater than ct, where c is the speed of light, vanishes as t goes to infinity as an inverse power of t., Comment: 21 pages

We do the spectral analysis of the Hamiltonian for the weak leptonic decay of the gauge bosons W+/-. Using Mourre theory, it is shown that the spectrum between the unique ground state and the first threshold is purely absolutely continuous. Neither sharp neutrino high energy cutoff nor infrared regularization are assumed., Comment: To appear in Ann. Henri Poincar\'e

We consider an atom interacting with the quantized electromagnetic field in the standard model of non-relativistic QED. The nucleus is supposed to be fixed. We prove smoothness of the resolvent and local decay of the photon dynamics for quantum states in a spectral interval I just above the ground state energy. Our results are uniform with respect to I. Their proofs are based on abstract Mourre's theory, a Mourre inequality established in [FGS1], Hardy-type estimates in Fock space, and a low-energy dyadic decomposition., Comment: 31 pages

We consider a non-relativistic electron interacting with a classical magnetic field pointing along the $x_3$-axis and with a quantized electromagnetic field. The system is translation invariant in the $x_3$-direction and we consider the reduced Hamiltonian $H(P_3)$ associated with the total momentum $P_3$ along the $x_3$-axis. For a fixed momentum $P_3$ sufficiently small, we prove that $H(P_3)$ has a ground state in the Fock representation if and only if $E'(P_3)=0$, where $P_3 \mapsto E'(P_3)$ is the derivative of the map $P_3 \mapsto E(P_3) = \inf \sigma (H(P_3))$. If $E'(P_3) \neq 0$, we obtain the existence of a ground state in a non-Fock representation. This result holds for sufficiently small values of the coupling constant.