Convergent expansions in non-relativistic qed: Analyticity of the ground state

We consider the ground state of an atom in the framework of non-relativistic qed. We show that the ground state as well as the ground state energy are analytic functions of the coupling constant which couples to the vector potential, under the assumption that the atomic Hamiltonian has a non-degenerate ground state. Moreover, we show that the corresponding expansion coefficients are precisely the coefficients of the associated Raleigh-Schrödinger series. As a corollary we obtain that in a scaling limit where the ultraviolet cutoff is of the order of the Rydberg energy the ground state and the ground state energy have convergent power series expansions in the fine structure constant α, with α dependent coefficients which are finite for α?0.     

The renormalized electron mass in non-relativistic quantum electrodynamics

This work addresses the problem of infrared mass renormalization for a non-relativistic electron minimally coupled to the quantized electromagnetic field (the standard, translationally invariant system of an electron in non-relativistic QED). We assume that the interaction of the electron with the quantized electromagnetic field is subject to an ultraviolet regularization and an infrared regularization parametrized by σ>0. For the value p=0 of the conserved total momentum of electron and photon field, bounds on the renormalized mass are established which are uniform in σ→0, and the existence of a ground state is proved. For |p|>0 sufficiently small, bounds on the renormalized mass are derived for any fixed σ>0. A key ingredient of our proofs is the operator-theoretic renormalization group based on the isospectral smooth Feshbach map. It provides an explicit, finite algorithm for determining the renormalized electron mass at p=0 to any given precision.     

Construction of the ground state in nonrelativistic QED by continuous flows

For a nonrelativistic hydrogen atom minimally coupled to the quantized radiation field we construct the ground state projection Pgs by a continuous approximation scheme as an alternative to the iteration scheme recently used by Fröhlich, Pizzo, and the first author [V. Bach, J. Fröhlich, A. Pizzo, Infrared-finite algorithms in QED: The groundstate of an atom interacting with the quantized radiation field, Comm. Math. Phys. (2006), doi: 10.1007/s00220-005-1478-3]. That is, we construct Pgs=limt→∞Pt as the limit of a continuously differentiable family (Pt)_t?0 of ground state projections of infrared regularized Hamiltonians Ht. Using the ODE solved by this family of projections, we show that the norm of their derivative is integrable in t which in turn yields the convergence of Pt by the fundamental theorem of calculus.     

Quantum group of orientation-preserving Riemannian isometries

We formulate a quantum group analogue of the group of orientation-preserving Riemannian isometries of a compact Riemannian spin manifold, more generally, of a (possibly R-twisted and of compact type) spectral triple. The main advantage of this formulation, which is directly in terms of the Dirac operator, is that it does not need the existence of any ‘good’ Laplacian as in our previous works on quantum isometry groups. Several interesting examples, including those coming from Rieffel-type deformation as well as the equivariant spectral triples on SU_μ(2) and are discussed.     

Quantum symmetry groups of Hilbert modules equipped with orthogonal filtrations

We define and show the existence of the quantum symmetry group of a Hilbert module equipped with an orthogonal filtration. Our construction unifies the constructions of Banica–Skalski?s quantum symmetry group of a C^?$C^{?}$-algebra equipped with an orthogonal filtration and Goswami?s quantum isometry group of an admissible spectral triple.     

The failure of spectral synthesis on some types of discrete Abelian groups

The problem of spectral synthesis on arbitrary Abelian groups is solved in the negative.     

Infrared renormalization in non-relativistic QED and scaling criticality

We consider a spin- electron in the framework of non-relativistic Quantum Electrodynamics (QED). Let denote the fiber Hamiltonian corresponding to the conserved total momentum of the electron and the photon field, regularized by a fixed ultraviolet cutoff in the interaction term, and an infrared regularization parametrized by 0<σ?1. Ultimately, our goal is to remove the latter by taking σ↘0. We prove that there exists a constant 0<α₀?1 independent of σ>0 such that for all and all values of the finestructure constant 0<α<α₀, there exists a ground state eigenvalue of multiplicity two at the bottom of the essential spectrum. Moreover, we prove that the renormalized electron mass satisfies , uniformly in σ?0, in units where the bare mass has the value 1, and we prove the existence of the renormalized mass in the limit σ↘0. Our analysis uses the isospectral renormalization group method of Bach, Fröhlich, Sigal introduced in [V. Bach, J. Fröhlich, I.M. Sigal, Quantum electrodynamics of confined non-relativistic particles, Adv. Math. 137 (2) (1998) 299-395; V. Bach, J. Fröhlich, I.M. Sigal, Renormalization group analysis of spectral problems in quantum field theory, Adv. Math. 137 (1998) 205-298] and further developed in [V. Bach, T. Chen, J. Fröhlich, I.M. Sigal, Smooth Feshbach map and operator-theoretic renormalization group methods, J. Funct. Anal. 203 (1) (2003) 44-92; V. Bach, T. Chen, J. Fröhlich, I.M. Sigal, The renormalized electron mass in non-relativistic QED, J. Funct. Anal. 243 (2) (2007) 426-535]. The limit σ↘0 determines a scaling-critical (or endpoint type) renormalization group problem, in which the interaction is strictly marginal (of scale-independent size). A main result of this paper is the development of a method that provides rigorous control of the renormalization of a strictly marginal quantum field theory characterized by a non-trivial scaling limit. The key ingredients entering this analysis include a hierarchy of exact algebraic cancelation identities exploiting the spatial and gauge symmetries of the model, and a combination of the isospectral renormalization group method with the strong induction principle.     

Smoothness of transition maps in singular perturbation problems with one fast variable

This paper deals with the smoothness of the transition map between two sections transverse to the fast flow of a singularly perturbed vector field (one fast, multiple slow directions). Orbits connecting both sections are canard orbits, i.e. they first move rapidly towards the attracting part of a critical surface, then travel a distance near this critical surface, even beyond the point where the orbit enters the repelling part of the critical surface, and finally repel away from the surface. We prove that the transition map is smooth. In a transcritical situation however, where orbits from an attracting part of one critical manifold follow the repelling part of another critical manifold, the smoothness of the transition map may be limited, due to resonance phenomena that are revealed by blowing up the turning point! We present a polynomial example in R³.     

Conjugacy classes in Weyl groups and q-W algebras

We define noncommutative deformations of algebras of regular functions on certain transversal slices to the set of conjugacy classes in an algebraic group G which play the role of Slodowy slices in algebraic group theory. The algebras called q-W algebras are labeled by (conjugacy classes of) elements s of the Weyl group of G. The algebra is a quantization of a Poisson structure defined on the corresponding transversal slice in G with the help of Poisson reduction of a Poisson bracket associated to a Poisson–Lie group G^? dual to a quasitriangular Poisson–Lie group. We also define a quantum group counterpart of the category of generalized Gelfand–Graev representations and establish an equivalence between this category and the category of representations of the corresponding q-W algebra. The algebras can be regarded as quantum group counterparts of W-algebras. However, in general they are not deformations of the usual W-algebras.     

An open problem on the optimality of an asymptotic solution to Duffing’s nonlinear oscillation problem

Based on the renormalization group method, Kirkinis (2012) [8] obtained an asymptotic solution to Duffing’s nonlinear oscillation problem. Kirkinis then asked if the asymptotic solution is optimal. In this paper, an affirmative answer to the open problem is given by means of the homotopy analysis method.     

Coorbits of Quantum Homogeneous Spaces

The notion of coorbits for spaces with quantum group actions is introduced. A space with a quantum group action is given by a pair of algebras: an associative algebra which is the analog of a classical topological space, and a Hopf algebra which is the analog of a classical topological group. The Hopf algebra acts on the associative algebra via a comodule structure mapping which is also an algebra homomorphism. For a space with a quantum group action, a coorbit is a pair of spaces given by the image and the kernel of an algebra homomorphism from the associative algebra to the Hopf algebra. The coorbits of several types of quantum homogeneous spaces are discussed. In the case when the associative algebra is the group algebra of a group and the Hopf algebra is a quotient of the group algebra, the connection between the set of coorbits and the character group is established.     

Quantum walks and the size of the graph

A continuous-time quantum walk is modelled using a graph. In this short paper, we provide lower bounds on the size of a graph that would allow for some quantum phenomena to occur. Among other things, we show that, in the adjacency matrix quantum walk model, the number of edges is bounded below by a cubic function on the eccentricity of a periodic vertex. This gives some idea on the shape of a graph that would admit periodicity or perfect state transfer. We also raise some extremal type of questions in the end that could lead to future research.     

Ground states for translationally invariant Pauli-Fierz models at zero momentum

We consider the translationally invariant Pauli-Fierz model describing a charged particle interacting with the electromagnetic field. We show under natural assumptions that the fiber Hamiltonian at zero momentum has a ground state.     

Portfolio symmetry and momentum

This paper presents a novel theoretical framework to model the evolution of a dynamic portfolio (i.e., a portfolio whose weights vary over time), considering a given investment policy. The framework is based on graph theory and the quantum probability. Embedding the dynamics of a portfolio into a graph, each node of the graph representing a plausible portfolio, we provide the probabilities for a dynamic portfolio to lie on different nodes of the graph, characterizing its optimality in terms of returns. The framework embeds cross-sectional phenomena, such as the momentum effect, in stochastic processes, using portfolios instead of individual stocks. We apply our methodology to an investment policy similar to the momentum strategy of Jegadeesh and Titman (1993). We find that the strategy symmetry is a source of momentum.     

The Cremmer-Gervais solution of the Yang-Baxter equation

A direct proof is given of the fact that the Cremmer-Gervais -matrix satisfies the (Quantum) Yang-Baxter equation

     

Eigenfunction expansions and spectral projections for isotropic elasticity outside an obstacle

We consider the operator −Δ−αgraddiv acting on an exterior domain Ω in Rn (with α>0 and n=2,3) subject to Dirichlet boundary conditions. The spectral resolution for the operator is written in terms of an expansion of generalized eigenfunctions.