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Register a new userResolvent smoothness and local decay at low energies for the standard model of non-relativistic QED
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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 Mourres theory, a Mourre inequality established in [FGS1], Hardy-type estimates in Fock space, and a low-energy dyadic decomposition.
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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.
It was recently shown [2] that the resolvent algebra of a non-relativistic Bose field determines a gauge invariant (particle number preserving) kinematical algebra of observables which is stable under the automorphic action of a large family of interacting dynamics involving pair potentials. In the present article, this observable algebra is extended to a field algebra by adding to it isometries, which transform as tensors under gauge transformations and induce particle number changing morphisms of the observables. Different morphisms are linked by intertwiners in the observable algebra. It is shown that such intertwiners also induce time translations of the morphisms. As a consequence, the field algebra is stable under the automorphic action of the interacting dynamics as well. These results establish a concrete C*-algebraic framework for interacting non-relativistic Bose systems in infinite space. It provides an adequate basis for studies of long range phenomena, such as phase transitions, stability properties of equilibrium states, condensates, and the breakdown of symmetries.
We consider the Landau Hamiltonian $H_0$, self-adjoint in $L^2({mathbb R^2})$, whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues $Lambda_q$, $q in {mathbb Z}_+$. We perturb $H_0$ by a non-local potential written as a bounded pseudo-differential operator ${rm Op}^{rm w}({mathcal V})$ with real-valued Weyl symbol ${mathcal V}$, such that ${rm Op}^{rm w}({mathcal V}) H_0^{-1}$ is compact. We study the spectral properties of the perturbed operator $H_{{mathcal V}} = H_0 + {rm Op}^{rm w}({mathcal V})$. First, we construct symbols ${mathcal V}$, possessing a suitable symmetry, such that the operator $H_{mathcal V}$ admits an explicit eigenbasis in $L^2({mathbb R^2})$, and calculate the corresponding eigenvalues. Moreover, for ${mathcal V}$ which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of $H_{mathcal V}$ adjoining any given $Lambda_q$. We find that the effective Hamiltonian in this context is the Toeplitz operator ${mathcal T}_q({mathcal V}) = p_q {rm Op}^{rm w}({mathcal V}) p_q$, where $p_q$ is the orthogonal projection onto ${rm Ker}(H_0 - Lambda_q I)$, and investigate its spectral asymptotics.
In the present paper we investigate the set $Sigma_J$ of all $J$-self-adjoint extensions of a symmetric operator $S$ with deficiency indices $<2,2>$ which commutes with a non-trivial fundamental symmetry $J$ of a Krein space $(mathfrak{H}, [cdot,cdot])$, SJ=JS. Our aim is to describe different types of $J$-self-adjoint extensions of $S$. One of our main results is the equivalence between the presence of $J$-self-adjoint extensions of $S$ with empty resolvent set and the commutation of $S$ with a Clifford algebra ${mathcal C}l_2(J,R)$, where $R$ is an additional fundamental symmetry with $JR=-RJ$. This enables one to construct the collection of operators $C_{chi,omega}$ realizing the property of stable $C$-symmetry for extensions $AinSigma_J$ directly in terms of ${mathcal C}l_2(J,R)$ and to parameterize the corresponding subset of extensions with stable $C$-symmetry. Such a situation occurs naturally in many applications, here we discuss the case of an indefinite Sturm-Liouville operator on the real line and a one dimensional Dirac operator with point interaction.
The structure of the gauge invariant (particle number preserving) C*-algebra generated by the resolvents of a non-relativistic Bose field is analyzed. It is shown to form a dense subalgebra of the bounded inverse limit of a system of approximately finite dimensional C*-algebras. Based on this observation, it is proven that the closure of the gauge invariant algebra is stable under the dynamics induced by Hamiltonians involving pair potentials. These facts allow to proceed to a description of interacting Bosons in terms of C*-dynamical systems. It is outlined how the present approach leads to simplifications in the construction of infinite bosonic states and sheds new light on topics in many body theory.
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