Spectral Analysis for Systems of Atoms and Molecules Coupled to the Quantized Radiation Field

We consider systems of static nuclei and electrons – atoms and molecules – coupled to the quantized radiation field. The interactions between electrons and the soft modes of the quantized electromagnetic field are described by minimal coupling, p→p−e A (x), where A(x) is the electromagnetic vector potential with an ultraviolet cutoff. If the interactions between the electrons and the quantized radiation field are turned off, the atom or molecule is assumed to have at least one bound state. We prove that, for sufficiently small values of the fine structure constant α, the interacting system has a ground state corresponding to the bottom of its energy spectrum. For an atom, we prove that its excited states above the ground state turn into metastable states whose life-times we estimate. Furthermore the energy spectrum is absolutely continuous, except, perhaps, in a small interval above the ground state energy and around the threshold energies of the atom or molecule. Received: 3 September 1998 / Accepted: 17 March 1999     

Ground State and Resonances in the Standard Model of the Non-Relativistic QED

We prove existence of a ground state and resonances in the standard model of the non-relativistic quantum electro-dynamics (QED). To this end we introduce a new canonical transformation of QED Hamiltonians and use the spectral renormalization group technique with a new choice of Banach spaces.     

Adiabatic Theorem without a Gap Condition

We prove the adiabatic theorem for quantum evolution without the traditional gap condition. All that this adiabatic theorem needs is a (piecewise) twice differentiable finite dimensional spectral projection. The result implies that the adiabatic theorem holds for the ground state of atoms in quantized radiation field. The general result we prove gives no information on the rate at which the adiabatic limit is approached. With additional spectral information one can also estimate this rate.     

Positive Commutators and the Spectrum¶of Pauli--Fierz Hamiltonian of Atoms and Molecules

In this paper we study the energy spectrum of the Pauli–Fierz Hamiltonian generating the dynamics of nonrelativistic electrons bound to static nuclei and interacting with the quantized radiation field. We show that, for sufficiently small values of the elementary electric charge, and under weaker conditions than those required in [3], the spectrum of this Hamiltonian is absolutely continuous, except possibly in small neighbourhoods of the ground state energy and the ionization thresholds. In particular, it is shown that (for a large range of energies) there are no stable excited eigenstates. The method used to prove these results relies on the positivity of the commutator between the Hamiltonian and a suitably modified dilatation generator on photon Fock space. Received: 10 April 1998 / Accepted: 12 April 1999     

Lifshitz Tails for a Class of Schrödinger Operators with Random Breather-Type Potential

We derive bounds on the integrated density of states for a class of Schrödinger operators with a random potential. The potential depends on a sequence of random variables, not necessarily in a linear way. An example of such a random Schrödinger operator is the breather model, as introduced by Combes, Hislop and Mourre. For these models, we show that the integrated density of states near the bottom of the spectrum behaves according to the so called Lifshitz asymptotics. This result can be used to prove Anderson localization in certain energy/disorder regimes.     

Anyons from strings

The Nambu-Goto string in a three-dimensional (3D) Minkowski spacetime is quantized preserving Lorentz invariance and parity. The spectrum of massive states contains anyons. An ambiguity in the ground state energy is resolved by the 3D N=1 Green-Schwarz superstring, which has massless ground states describing a dilaton and dilatino, and first-excited states of spin 1/4.     

Ground State Energy of Large Atoms in a Self-Generated Magnetic Field

We consider a large atom with nuclear charge Z described by non-relativistic quantum mechanics with classical or quantized electromagnetic field. We prove that the absolute ground state energy, allowing for minimizing over all possible self-generated electromagnetic fields, is given by the non-magnetic Thomas-Fermi theory to leading order in the simultaneous Z → ∞, α → 0 limit if Z α ² ≤ κ for some universal κ, where α is the fine structure constant.     

Quantization of a self-interacting maximally charged string

We discuss the quantization of a self-interacting string consisting of maximally charged matter. We construct the Hamiltonian in the non-relativistic limit by expanding around a static solution of the Einstein–Maxwell field equations. Conformal symmetry is broken on the worldsheet, but a subgroup of the conformal group acts as the gauge group of the theory. Thus, the Faddeev–Popov quantization procedure of fixing the gauge is applicable. We calculate the Hamiltonian and show that, if properly quantized, the system possesses a well-defined ground state and the spacing of its energy levels is of order the Planck mass. This generalizes earlier results on a system of maximally charged black holes to the case of continuous matter distributions.     

Spectral Theory of Massless Pauli-Fierz Models

We study the spectral theory of massless Pauli-Fierz models using an extension of the Mourre method. We prove the local finiteness of point spectrum and a limiting absorption principle away from the eigenvalues for an arbitrary coupling constant. In addition we show that the expectation value of the number operator is finite on all eigenvectors.Supported by Carlsbergfondet     

Second Order Perturbation Theory for Embedded Eigenvalues

We study second order perturbation theory for embedded eigenvalues of an abstract class of self-adjoint operators. Using an extension of the Mourre theory, under assumptions on the regularity of bound states with respect to a conjugate operator, we prove upper semicontinuity of the point spectrum and establish the Fermi Golden Rule criterion. Our results apply to massless Pauli-Fierz Hamiltonians for arbitrary coupling.     

Coulomb system equivalent to the energy spectrum of the Calogero-Sutherland-Moser (CSM) model

The purpose of this paper is to prove an equivalence between the energy spectrum of the CSM model and the electrostatic energy of a one-dimensional lattice of quantized point charges interacting via Coulomb potential with Dirichlet boundary conditions. This paper is dedicated to B. Jancovici in honor of his 65th birthday.     

Destruction of classical cantori in the quantum Frenkel-Kontorova model

In this paper we investigate the properties of the quantized discrete Frenkel Kontorova model. The structure of the ground state is numerically analyzed by means of the Metropolis algorithm; special attention is given to the effects of quantization on the Cantori structure of the classical ground state. These quantum effects produce a new structure which can be approximately described by using a sawtooth map instead of the Standard map. The dependence of the quantum energy on temperature is also investigated and discussed in connection with these structural modifications.     

Hyperfine Splitting in Non-relativistic QED: Uniqueness of the Dressed Hydrogen Atom Ground State

We consider a free hydrogen atom composed of a spin- ${\frac{1}{2}}$ nucleus and a spin- ${\frac{1}{2}}$ 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.     

Minimal Photon Velocity Bounds in Non-relativistic Quantum Electrodynamics

We consider non-relativistic quantum particle systems, such as atoms and molecules, coupled to the quantized electromagnetic field. We prove several photon velocity bounds for total energies below the ionization threshold. We also consider phonons coupled to such particle systems and prove velocity bounds for them as well.     

Derivation of the Gross-Pitaevskii Equation for Rotating Bose Gases

We prove that the Gross-Pitaevskii equation correctly describes the ground state energy and corresponding one-particle density matrix of rotating, dilute, trapped Bose gases with repulsive two-body interactions. We also show that there is 100% Bose-Einstein condensation. While a proof that the GP equation correctly describes non-rotating or slowly rotating gases was known for some time, the rapidly rotating case was unclear because the Bose (i.e., symmetric) ground state is not the lowest eigenstate of the Hamiltonian in this case. We have been able to overcome this difficulty with the aid of coherent states. Our proof also conceptually simplifies the previous proof for the slowly rotating case. In the case of axially symmetric traps, our results show that the appearance of quantized vortices causes spontaneous symmetry breaking in the ground state.     

Criterion for a unique non-relativistic quark model

We have calculated the mean square charge radii of the neutron and proton, and compared them with the experimental values, to construct a unique non-relativistic quark potential model. It is shown for the first time that, contrary to general belief, it is possible to reproduce simultaneously the baryon mass spectrum and the electromagnetic sizes of neutron and proton using a single potential model. It was found necessary to add admixtures of excited states of the nucleon in the unperturbed ground state wavefunctions.     

A Second Quantized Approach to the Rabi Problem

In the present work, the Rabi Problem, involving the response of a spin 1/2 particle subjected to a magnetic field, is considered in a second quantized approach. In this concrete physical scenario, we show that the second quantization procedure can be applied directly in a non-covariant theory. The proposed development explicits not only the relation between the full quantum treatment of the problem and the semiclassical Rabi model, but also the connection of these approaches with the Jaynes-Cummings model. The consistency of the method is checked in the semiclassical limit. The treatment is then extended to the matter component of the Rabi problem so that the Schrödinger equation is directly quantized. Considering the spinorial field, the appearance of a negative energy sector implies a specific identification between Schrödinger’s and Maxwell’s theories. The generalized theory is consistent, strictly quantum and non-relativistic.     

Light meson spectrum and their leptonic decay widths in the frame work of constituent quark models

The phenomenological non-relativistic quark model has been employed to obtain the ground state masses of light vector mesons and their radially excited states and their decay widths.The full hamiltonian used in the investigation has kinetic energy,the confinement potential and the one-gluon-exchange potential.A good agreement is obtained with the experimental masses and their leptonic decay widths.     

Absence of Ground States for a Class of Translation Invariant Models of Non-relativistic QED

We consider a class of translation invariant models of non-relativistic QED with net charge. Under certain natural assumptions we prove that ground states do not exist in the Fock space.