Numerically exact quantum dynamics for indistinguishable particles: The multilayer multiconfiguration timedependent Hartree theory in second quantization representation Extensions of representations of the CAR algebra to the Cuntz algebra O 2 -the Fock and the infinite wedgeIn its original formulation, Lieb's variational principle holds for fermion systems with purely repulsive pair interactions. As a generalization we prove for both fermion and boson systems with semi-bounded Hamiltonian that the infimum of the energy over quasifree states coincides with the infimum over pure quasifree states. In particular, the Hamiltonian is not assumed to preserve the number of particles. To shed light on the relation between our result and the usual formulation of Lieb's variational principle in terms of one-particle density matrices, we also include a characterization of pure quasifree states by means of their generalized one-particle density matrices. C 2014 AIP Publishing LLC. [http://dx.

According to Anderson's orthogonality catastrophe, the overlap of the N -particle ground states of a free Fermi gas with and without an (electric) potential decays in the thermodynamic limit. For the finite one-dimensional system various boundary conditions are employed. Unlike the usual setup the perturbation is introduced by a magnetic (vector) potential. Although such a magnetic field can be gauged away in one spatial dimension there is a significant and interesting effect on the overlap caused by the phases. We study the leading asymptotics of the overlap of the two ground states and the two-term asymptotics of the difference of the ground-state energies. In the case of periodic boundary conditions our main result on the overlap is based upon a well-known asymptotic expansion by Fisher and Hartwig on Toeplitz determinants with a discontinuous symbol. In the case of Dirichlet boundary conditions no such result is known to us and we only provide an upper bound on the overlap, presumably of the right asymptotic order.

It is shown in this paper that the G-Condition and the P-Condition from representability imply the fermion correlation estimate from [1] which, in turn, is known to yield a nontrivial bound on the accuracy of the Hartree-Fock approximation for large Coulomb systems.

Representability conditions on the one-and two-particle density matrix for fermion systems are formulated by means of Grassmann integrals. A positivity condition for a certain kind of Grassmann integral is established which, in turn, induces the wellknown G-, P-and Q-Conditions of quantum chemistry by an appropriate choice of the integrand. Similarly, the T 1 -and T 2 -Conditions are derived. Furthermore, quasifree Grassmann states are introduced and, for every operator γ ∈ H ⊕ H with 0 ≤ γ ≤ 1, the existence of a unique quasifree Grassmann state whose one-particle density matrix is γ is shown.

We consider a reduced two-channel model of an atom consisting of a quantum dot coupled to an open scattering channel described by a three-dimensional Laplacian. We are interested in the survival probability of a bound state when the dot energy varies smoothly and adiabatically in time. The initial state corresponds to a discrete eigenvalue which dives into the continuous spectrum and re-emerges from it as the dot energy is varied in time and finally returns to its initial value. Our main result is that for a large class of couplings, the survival probability of this bound state vanishes in the adiabatic limit. At the end of the paper we present a short outlook on how our method may be extended to cover other classes of Hamiltonians; details will be given elsewhere.the existence of the so-called spontaneous (adiabatic) pair creation in linear quantum electrodynamics [10,11,15], as well as for memory effects in quantum mesoscopic transport [2,3,4,9].Turning this heuristics into a mathematical statement proved to be a hard problem and boiled down to a proof of various aspects of the adiabatic theorem in the case where the eigenvalue hits the threshold of the continuous spectrum. Accordingly, the existence results are very limited and the proofs are rather technical and often need further assumptions [3,10,11].Our main result states that the survival probability vanishes in the adiabatic limit, i.e. the adiabatic theorem breaks down, for a large class of couplings between the quantum dot and the open channel, when the bound state dives into the continuous spectrum during the adiabatic tuning of E. In addition, a detailed spectral analysis of the model is given and a 'threshold adiabatic theorem' is proved.Our model is considerably simpler than the one in [11] and allows for rather straightforward dispersive estimates, while the threshold analysis is self-contained. In spite of the relative simplicity of the model, our method is quite robust and may be generalized to cover a larger class of operators. In Section 7 we present a short outlook on the Dirac and N-body Schrödinger case with N ≥ 2. Details will be given elsewhere.Below we present the setting, formulate the results, and comment on them. Sections 2-6 contain the proofs of the statements in the main theorem.