Michael O'Carroll

Asymptotic completeness for multi-particle Schroedinger Hamiltonians with weak potentials

AbstractWe show that the non-relativistic quantum mechanicaln-body HamiltoniansT(k)=T+kV andT, the free particle Hamiltonian, are unitarily equivalent in the center of mass system, i.e.,T(k)=W±(k)TW±(k)−1 fork sufficiently small and real.

Transfer Matrix Spectrum and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

V = \sum\limits_i {V_i }

2+1

, a sum ofn(n−1)/2 real pair potentials,Vi, depending on the relative coordinatexi∈R3 of the pairi, whereVi is required to behave like |xi|− 2 −ε as |xi|→∞ and like |xi|− 2 +ε as |xi|→0.T(k) is the self-adjoint operator associated with the form sumT+kV. There are no smoothness requirements imposed on theVi. Furthermore

A representation for the generating and correlation functions in the block field renormalization group formalism and asymptotic freedom

W_ \pm (k) = \mathop {s - \lim }\limits_{t \to \pm \infty } e^{iT(k)t} e^{ - iTt}

Low temperature properties of the hierarchical classical vector model

are the wave operators of time dependent scattering theory and are unitary. This result gives a quantitative form of the intuitive argument based on the Heisenberg uncertainty principle that a certain minimum potential well depth and range is needed before a bound state can be formed. This is the best possible long range behavior in the sense that ifkVi≦Ci|xi|−b, 0Ri(0

Transfer Matrix Spectrum for Lattice Classical O(N) Ferromagnetic Spin Systems at High Temperature

Block renormalization group transformations (RGT) for lattice and continuum Euclidean Fermions in d dimensions are developed using Fermionic integrals with exponential and “δ-function” weight functions. For the free field the sequence of actionsDk generated by the RGT from D, the Dirac operator, are shown to have exponential decay; uniform ink, after rescaling to the unit lattice. It is shown that the two-point functionD−1 admits a simple telescopic sum decomposition into fluctuation two-point functions which for the exponential weight RGT have exponential decay. Contrary to RG intuition the sequence of rescaled actions corresponding to the “δ-function” RGT do not have uniform exponential decay and we give examples of initial actions in one dimension where this phenomena occurs for the exponenential weight RGT also.