Thomas Chen

Smooth Feshbach map and operator-theoretic renormalization group methods

Abstract A new variant of the isospectral Feshbach map defined on operators in Hilbert space is presented. It is constructed with the help of a smooth partition of unity, instead of projections, and is therefore called smooth Feshbach map . It is an effective tool in spectral and singular perturbation theory. As an illustration of its power, a novel operator-theoretic renormalization group method is described and applied to analyze a general class of Hamiltonians on Fock space. The main advantage of the new renormalization group method over its predecessors is its technical simplicity, which it owes to the use of the smooth Feshbach map.

Unconditional Uniqueness for the Cubic Gross‐Pitaevskii Hierarchy via Quantum de Finetti

We present a new, simpler proof of the unconditional uniqueness of solutions to the cubic Gross-Pitaevskii hierarchy in . One of the main tools in our analysis is the quantum de Finetti theorem. Our uniqueness result is equivalent to the one established in the celebrated works of Erdős, Schlein, and Yau.

Infraparticle Scattering States in Non-Relativistic QED: I. The Bloch-Nordsieck Paradigm

We construct infraparticle scattering states for Compton scattering in the standard model of non-relativistic QED. In our construction, an infrared cutoff initially introduced to regularize the model is removed completely. We rigorously establish the properties of infraparticle scattering theory predicted in the classic work of Bloch and Nordsieck from the 1930’s, Faddeev and Kulish, and others. Our results represent a basic step towards solving the infrared problem in (non-relativistic) QED.

Achieving optimal growth through product feedback inhibition in metabolism.

Recent evidence suggests that the metabolism of some organisms, such as Escherichia coli, is remarkably efficient, producing close to the maximum amount of biomass per unit of nutrient consumed. This observation raises the question of what regulatory mechanisms enable such efficiency. Here, we propose that simple product-feedback inhibition by itself is capable of leading to such optimality. We analyze several representative metabolic modules—starting from a linear pathway and advancing to a bidirectional pathway and metabolic cycle, and finally to integration of two different nutrient inputs. In each case, our mathematical analysis shows that product-feedback inhibition is not only homeostatic but also, with appropriate feedback connections, can minimize futile cycling and optimize fluxes. However, the effectiveness of simple product-feedback inhibition comes at the cost of high levels of some metabolite pools, potentially associated with toxicity and osmotic imbalance. These large metabolite pool sizes can be restricted if feedback inhibition is ultrasensitive. Indeed, the multi-layer regulation of metabolism by control of enzyme expression, enzyme covalent modification, and allostery is expected to result in such ultrasensitive feedbacks. To experimentally test whether the qualitative predictions from our analysis of feedback inhibition apply to metabolic modules beyond linear pathways, we examine the case of nitrogen assimilation in E. coli, which involves both nutrient integration and a metabolic cycle. We find that the feedback regulation scheme suggested by our mathematical analysis closely aligns with the actual regulation of the network and is sufficient to explain much of the dynamical behavior of relevant metabolite pool sizes in nutrient-switching experiments.

Infraparticle scattering states in nonrelativistic quantum electrodynamics. II. Mass shell properties

We study the infrared problem in the usual model of quantum electrodynamics with nonrelativistic matter. We prove spectral and regularity properties characterizing the mass shell of an electron and one-electron infraparticle states of this model. Our results are crucial for the construction of infraparticle scattering states, which are treated in a separate paper.

The increase of binding energy and enhanced binding in nonrelativistic QED

We consider a Pauli–Fierz Hamiltonian for a particle coupled to a photon field. We discuss the effects of the increase of the binding energy and enhanced binding through coupling to a photon field, and prove that both effects are the results of the existence of the ground state of the self-energy operator with total momentum P=0.

Energy conservation and blowup of solutions for focusing Gross-Pitaevskii hierarchies

Abstract We consider solutions of the focusing cubic and quintic Gross–Pitaevskii (GP) hierarchies. We identify an observable corresponding to the average energy per particle, and we prove that it is a conserved quantity. We prove that all solutions to the focusing GP hierarchy at the L 2 -critical or L 2 -supercritical level blow up in finite time if the energy per particle in the initial condition is negative. Our results do not assume any factorization of the initial data.

Binding conditions for atomic N-electron systems in non-relativistic QED

Abstract. We examine the binding conditions for atoms in non-relativistic QED, and prove that removing one electron from an atom requires a positive energy. As an application, we establish the existence of a ground state for the Helium atom.

A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies

We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in d ≥ 1 dimensions, for focusing and defocusing interactions. We present a new proof of existence of solutions that does not require the a priori bound on the spacetime norm, which was introduced in the work of Klainerman and Machedon, [19], and used in our earlier work, [6].

Higher Order Energy Conservation and Global Well-Posedness of Solutions for Gross-Pitaevskii Hierarchies

We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in d dimensions, for focusing and defocusing interactions. We introduce new higher order energy functionals and prove that they are conserved for solutions of energy subcritical defocusing, and L 2 subcritical (de)focusing GP hierarchies, in spaces also used by Erdös, Schlein and Yau. By use of this tool, we prove a priori H 1 bounds for positive semidefinite solutions in those spaces. Moreover, we obtain global well-posedness results for positive semidefinite solutions in the spaces studied in the works of Klainerman and Machedon. As part of our analysis, we prove generalizations of Sobolev and Gagliardo-Nirenberg inequalities for density matrices.