Quantum Diffusion for the Anderson Model in the Scaling Limit

We consider random Schr?dinger equations on for d ≥ 3 with identically distributed random potential. Denote by λ the coupling constant and ψ_t the solution with initial data ψ₀. The space and time variables scale as with 0 < κ < κ₀(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψ_t converges weakly to a solution of a heat equation in the space variable x for arbitrary L ² initial data. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum υ. This work is an extension to the lattice case of our previous result in the continuum [8,9]. Due to the non-convexity of the level surfaces of the dispersion relation, the estimates of several Feynman graphs are more involved. Submitted: April 18, 2006. Accepted: October 12, 2006. László Erdős: Partially supported by NSF grant DMS-0200235 and EU-IHP Network ‘Analysis and Quantum’ HPRN-CT-2002-0027. Manfred Salmhofer: Partially supported by DFG grant Sa 1362/1-1 and an ESI senior research fellowship. Horng-Tzer Yau: Partially supported by NSF grant DMS-0307295 and MacArthur Fellowship.     

Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams

We consider random Schrödinger equations on \({\mathbb{R}^{d}}\) for d≥ 3 with a homogeneous Anderson-Poisson type random potential. Denote by λ the coupling constant and ψ _t the solution with initial data ψ₀. The space and time variables scale as \({x\sim \lambda^{-2 -\kappa/2}, t \sim \lambda^{-2 -\kappa}}\) with 0 < κ <  κ₀(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψ _t converges weakly to the solution of a heat equation in the space variable x for arbitrary L ² initial data. The proof is based on a rigorous analysis of Feynman diagrams. In the companion paper [10] the analysis of the non-repetition diagrams was presented. In this paper we complete the proof by estimating the recollision diagrams and showing that the main terms, i.e. the ladder diagrams with renormalized propagator, converge to the heat equation.     

Large Matrix Computations on Vector Computers

Preconditionings have proved to be a powerful technique for accelerating the rate of convergence of an iterative method. This paper, which is concerned with the conjugate gradient algorithm for large matrix computations, investigates an approximate polynomial preconditioning strategy. The method is particularly attractive for implementation on vector computers.     

Non-Existence of Black Hole Solutions¶for a Spherically Symmetric, Static Einstein–Dirac–Maxwell System

We consider for j=?, … a spherically symmetric, static system of (2j+1) Dirac particles, each having total angular momentum j. The Dirac particles interact via a classical gravitational and electromagnetic field. The Einstein–Dirac–Maxwell equations for this system are derived. It is shown that, under weak regularity conditions on the form of the horizon, the only black hole solutions of the EDM equations are the Reissner–Nordstr?m solutions. In other words, the spinors must vanish identically. Applied to the gravitational collapse of a “cloud” of spin-?-particles to a black hole, our result indicates that the Dirac particles must eventually disappear inside the event horizon. Received: 2 November 1998 / Accepted: 23 February 1999     

Nonexistence of time‐periodic solutions of the Dirac equation in an axisymmetric black hole geometry

We prove that in the nonextreme Kerr‐Newman black hole geometry, the Dirac equation has no normalizable, time‐periodic solutions. A key tool is Chan‐drasekhar's separation of the Dirac equation in this geometry. A similar nonexistence theorem is established in a more general class of stationary, axisymmetric metrics in which the Dirac equation is known to be separable. These results indicate that, in contrast to the classical situation of massive particle orbits, a quantum mechanical Dirac particle must either disappear into the black hole or escape to infinity. © 2000 John Wiley & Sons, Inc.     

Morphology and glass transition behavior of polycarbonate–phenoxy system: Effects of trans-reactions in domain interface regions

Effects of trans reactions on the morphology, glass transition, and phase behavior in a classical blend system of a poly(hydroxyl ether bisphenol-A) (phenoxy) with bisphenol-A polycarbonate (PC) were investigated by differential scanning calorimetry (DSC) and optical microscopy. Although two T_gs were observed in the as-prepared PC/phenoxy blends, an apparently single, but broadened, T_g was found in the blends after heating at high temperatures, typically 200–250°C for short times. The optical microscopy results indicated that same scales of heterogeneity did exist in post-heated PC/phenoxy blends as well as unheated blends. Explanations were provided. After heating-induced interchange reactions ( OH and carbonate), randomly linked polymer chains might form at the numerous interfaces of the mutually occluded/included micro-domains. The majority of the chains in the micro-domains are forced to relax in coordinated motion modes after heating, thus showing a single T_g. A mechanism of trans reactions in interfacial regions was briefly discussed in supplement to earlier reports in the literature. © 1997 John Wiley & Sons, Inc.