Quantitative Estimates on the Binding Energy for Hydrogen in Non-Relativistic QED. II. The spin case

The hydrogen binding energy in the Pauli-Fierz model with the spin Zeeman term is determined up to the order alpha cube, where alpha denotes the fine-structure constant

Spectral theory for a mathematical model of the weak interaction: The decay of the intermediate vector bosons W+/-, II

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\'

Non-analyticity of the groud state energy of the Hamiltonian for Hydrogen atom in non-relativistic QED

We derive the ground state energy up to the fourth order in the fine structure constant $\alpha$ for the translation invariant Pauli-Fierz Hamiltonian for a spinless electron coupled to the quantized radiation field. As a consequence, we obtain the non-analyticity of the ground state energy of the Pauli-Fierz operator for a single particle in the Coulomb field of a nucleus

Spectral theory for a mathematical model of the weak interaction: The decay of the intermediate vector bosons W+/-. I

We consider a Hamiltonian with cutoffs describing the weak decay of spin one massive bosons into the full family of leptons. The Hamiltonian is a self-adjoint operator in an appropriate Fock space with a unique ground state. We prove a Mourre estimate and a limiting absorption principle above the ground state energy and below the first threshold for a sufficiently small coupling constant. As a corollary, we prove absence of eigenvalues and absolute continuity of the energy spectrum in the same spectral interval.Comment: Correction of minor misprint

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as $\lambda \to \infty$, $\dim (\sigma(H_\lambda)) \cdot \log \lambda$ converges to an explicit constant ($\approx 0.88137$). We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schr\"odinger dynamics generated by the Fibonacci Hamiltonian.Comment: 23 page

Quantitative estimates on the Hydrogen ground state energy in non-relativistic QED

In this paper, we determine the exact expression for the hydrogen binding energy in the Pauli-Fierz model up to the order $O(\alpha^5\log\alpha^{-1})$, where $\alpha$ denotes the finestructure constant, and prove rigorous bounds on the remainder term of the order $o(\alpha^5\log\alpha^{-1})$. As a consequence, we prove that the binding energy is not a real analytic function of $\alpha$, and verify the existence of logarithmic corrections to the expansion of the ground state energy in powers of $\alpha$, as conjectured in the recent literature.Comment: AMS Latex, 51 page

Exponential localization of hydrogen-like atoms in relativistic quantum electrodynamics

We consider two different models of a hydrogenic atom in a quantized electromagnetic field that treat the electron relativistically. The first one is a no-pair model in the free picture, the second one is given by the semi-relativistic Pauli-Fierz Hamiltonian. We prove that the no-pair operator is semi-bounded below and that its spectral subspaces corresponding to energies below the ionization threshold are exponentially localized. Both results hold true, for arbitrary values of the fine-structure constant, $e^2$, and the ultra-violet cut-off, $\Lambda$, and for all nuclear charges less than the critical charge without radiation field, $Z_c=e^{-2}2/(2/\pi+\pi/2)$. We obtain similar results for the semi-relativistic Pauli-Fierz operator, again for all values of $e^2$ and $\Lambda$ and for nuclear charges less than $e^{-2}2/\pi$.Comment: 37 page

Quantum Return Probability for Substitution Potentials

We propose an effective exponent ruling the algebraic decay of the average quantum return probability for discrete Schrodinger operators. We compute it for some non-periodic substitution potentials with different degrees of randomness, and do not find a complete qualitative agreement with the spectral type of the substitution sequences themselves, i.e., more random the sequence smaller such exponent.Comment: Latex, 13 pages, 6 figures; to be published in Journal of Physics

Existence and uniqueness of the integrated density of states for Schr\"odinger operators with magnetic fields and unbounded random potentials

The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schr\"odinger operator with a constant magnetic field and a random potential which may be unbounded from above and from below. For an ergodic random potential satisfying a simple moment condition, we give a detailed proof that the infinite-volume limits of spatial eigenvalue concentrations of finite-volume operators with different boundary conditions exist almost surely. Since all these limits are shown to coincide with the expectation of the trace of the spatially localized spectral family of the infinite-volume operator, the integrated density of states is almost surely non-random and independent of the chosen boundary condition. Our proof of the independence of the boundary condition builds on and generalizes certain results by S. Doi, A. Iwatsuka and T. Mine [Math. Z. {\bf 237} (2001) 335-371] and S. Nakamura [J. Funct. Anal. {\bf 173} (2001) 136-152].Comment: This paper is a revised version of the first part of the first version of math-ph/0010013. For a revised version of the second part, see math-ph/0105046. To appear in Reviews in Mathematical Physic