We study the renormalization invariant trajectory of the $\phi^4$-perturbation of the free field fixed point in the hierarchical approximation. We parametrize it by a running $\phi^4$-coupling $g$ with linear step $\beta$-function. We rigorously control the non-perturbative corrections to finite order approximants from double perturbation theory in $g$ and $g^2\ln(g)$. The construction uses a contraction mapping for the extended renormalization group composed of a hierarchical block spin transformation with a flow of $g$.
In this article we prove several reciprocity theorems for some infinite-dimensional dual pairs of representations on Bargmann-Segal-Fock spaces.
We consider the nonlinear Schrodinger equation with a pure power repulsive nonlinearity on Schwarzschild manifolds. Equations of this type arise when a nonlinear wave equation on a Schwarzschild manifold is written in Hamiltonian form, cf. [2], [10]. For radial solutions with sufficiently localized initial data, we obtain global existence, L-p estimates, and the existence and asymptotic completeness of the wave operators. Our approach is based on a dilation identity and global space-time estimates.
We study the high energy limit of the Faddeev Scattering Amplitude for the Dirac operator associated with a potential Q. We prove that the Fourier transform of the potential and the limit of the scattering amplitude are related by an integral equation. Finally we apply these results to reconstruct the potential module a gauge transformation.
Considering the nonrelativistic Schrodinger operator for homonuclear diatomic molecular ions within the clamped nuclei approximation, we study the stability problem for increasing ratio Z/N of nuclear charge Z to number N of electrons. In particular, we derive improved bounds on the critical parameters that imply instability (upper bounds on the nuclear charge, or, equivalently, lower bounds on the number of electrons), viz. parameters that lead to dissociation of the molecular system into atomic fragments. The principal qualitative advantange of our estimates is the inclusion of electronic correlation, i.e., taking into account the effect of electron-electron repulsion on the molecular bond. Comparing; our rigorous results with empirical or computed data, we formulate a conjecture that should quantify the actual stability behaviour of realistic molecular species.
