Localization for off-diagonal disorder and for continuous Schrödinger operators

We extend the proof of localization by Delyon, Lévy, and Souillard to accommodate the Anderson model with off-diagonal disorder and the continuous Schrödinger equation with a random potential.Work supported in part by Nato under Nato Grant # 346/84Research partially supported by USNSF under grant DMS 84-16049     

Removable singularities of coupled Yang-Mills fields inR 3

We consider isolated point singularities of the coupled Yang-Mills equations inR ³. Under appropriate conditions on the curvature and the Higgs field, a removable singularity theorem is proved.Research supported in part by NSF grant MCS81-03403 and by Sonderforschungsbereich 72, University of BonnResearch supported by Sonderforschungsbereich 72, University of Bonn     

The infinite-volume ground state of the Lennard-Jones potential

We consider a finite chain of particles in one dimension, interacting through the Lennard-Jones potential. We prove the ground state is unique, and approaches uniform spacing in the infinite-particle limit.Research supported in part by NSF grant MCS 78-01520.     

Rigorous entropy-energy arguments

We present a method for making rigorous various arguments which predict that certain situations are unstable because of a balance of energy vs. entropy. As applications, we give yet another proof that the two-dimensional plane rotor has no spontaneous magnetization and we make rigorous Thouless' arguments on the one-dimensional Ising model with couplingJ/n ².on leave from Departments of Mathematics and Physics, Princeton University. Research partially supported by U.S.N.S.F. under grant No. MCS-78-01885.Danforth Fellow. Research partially supported by U.S.N.S.F. under grant No. PHY 78-23952.     

Classical and quantum-mechanical systems of Toda lattice type. I

The structure of the commutant of Laplace operators in the enveloping and Poisson algebra of certain generalized ax +b groups leads (in this article) to a determination of classical and quantum mechanical first integrals to generalized periodic and non-periodic Toda lattices. Certain new Hamiltonian systems of Toda lattice type are also shown to fit in this framework. Finite dimensional Lax forms for the (periodic) Toda lattices are given generalizing results of Flaschke.Research partially supported by NSF grant MCS 79-03223Research partially supported by NSF grant MCS 79-03153     

The symmetry of ground states under perturbation

We consider a two-body potential which has only periodic ground states and prove that it can be perturbed, by an arbitrarily small perturbation, so as to have only aperiodic ground states.Research supported in part by NSF grant MCS 78-01520-A01.     

Harmonic analysis on SL(2,R) and smoothness of the density of states in the one-dimensional Anderson model

We consider infinite Jacobi matrices with ones off-diagonal, and independent identically distributed random variables with distributionF(v)dv on-diagonal. IfF has compact support and lies in some Sobolev spaceL ¹ , then we prove that the integrated density of states,k(E), isC inE.Research partially supported by USNSF under Grant MCS-81-20833Research partially supported by USNSF under Grant MCS-82-01766A01     

The ground state for sticky disks

It is proven that the ground state of the two-dimensional sticky potential is the triangular lattice.Research supported in part by NSF grant MCS 78-01520-A01.     

Support theorems for random Schrödinger operators

In the multi-dimensional case it is shown that the increase of the topological support of the probability measure describing the randomness of potentials implies the increase of the spectrum. In the one-dimensional case the converse statement for the absolutely continuous spectrum is valid. Especially the spectrum (in general dimension) and the absolutely continuous spectrum (in one-dimension) are determined only by the topological support of the random potentials.Research partially supported by the USNSF under grant MCS-81-20833     

Schrödinger operators with magnetic fields

We prove a large number of results about atoms in constant magnetic field including (i) Asymptotic formula for the ground state energy of Hydrogen in large field, (ii) Proof that the ground state of Hydrogen in an arbitrary constant field hasL _z = 0 and of the monotonicity of the binding energy as a function ofB, (iii) Borel summability of Zeeman series in arbitrary atoms, (iv) Dilation analyticity for arbitrary atoms with infinite nuclear mass, and (v) Proof that every once negatively charged ion has infinitely many bound states in non-zero magnetic field with estimates of the binding energy for smallB and largeL _z .On leave at Department of Mathematics, California Institute of Technology, Pasadena, CA 91125 for 1980/81Supported by USNSF Grant MCS-78-00101Supported by USNSF Grant MCS-78-01885; on leave at Department of Mathematics, California Institute of Technology, Pasadena, CA 91125 for 1980/81     

Pure states as a dual object forC*-algebras

We consider the set of pure states of aC*-algebra as a uniform space equipped with transition probabilities and orientation, and show that the pure states with this structure determine theC*-algebra up to *-isomorphism.Partially supported by NSF grant MCS78-02455     

Correlation inequalities and contour estimates

We give a simple estimate on the probability of contours in classical ferromagnetic spin systems, based on Griffiths' or Ginibre's correlation inequalities. This includes quite general one- and two-component spin models. Some extension also holds for alln-component anisotropic or isotropic rotators.Supported by NSF grant No. MCS78-01885.On leave from: Institut de Physique Théorique, Université de Louvain, Belgium.Supported by NSF grant No. PHY78-15920.     

Lifschitz tails for the Anderson model

We present, in an expository way, an elementary rigorous proof (patterned after an argument of Kirsch-Martinelli) that the Anderson model has Lifschitz tails in very great generality.Research partially supported by USNSF Grant No. MCS-81-20833.     

Asymptotic Completeness for <Emphasis Type="Italic">N</Emphasis>-Body Quantum Systems with Long-Range Interactions in a Time-Periodic Electric Field

We show the asymptotic completeness for N-body quantum systems with long-range interactions in a time-periodic electric field whose mean in time is non-zero, where N ≥ 2. One of the main ingredients of this paper is to give some propagation estimates for physical propagators generated by time-periodic Hamiltonians which govern the systems under consideration. Research partially supported by the Grant-in-Aid for Young Scientists of MEXT #17740078.     

Two-Dimensional Critical Percolation: The Full Scaling Limit

We use SLE ₆ paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice – that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.Research partially supported by a Marie Curie Intra-European Fellowship under contract MEIF-CT-2003-500740 and by a Veni grant of the Dutch Organization for Scientific Research (NWO).Research partially supported by the U.S. NSF under grant DMS-01-04278.     

The positivity of the pressure in Thomas Fermi theory

We prove the positivity of the pressure and compressibility for neutral systems in the Thomas-Fermi theory of molecules. Our results include some new properties of the Thomas-Fermi potential and a proof that the kinetic energy is superadditive.Work partially supported by U.S. National Science Foundation grant MCS 75-21684 A02On leave from Department of Physics, Universidad de Chile, Santiago, Chile     

The Hamiltonian structure of general relativistic perfect fluids

We show that the evolution equations for a perfect fluid coupled to general relativity in a general lapse and shift, are Hamiltonian relative to a certain Poisson structure. For the fluid variables, a Lie-Poisson structure associated to the dual of a semi-direct product Lie algebra is used, while the bracket for the gravitational variables has the usual canonical symplectic structure. The evolution is governed by a Hamiltonian which is equivalent to that obtained from a canonical analysis. The relationship of our Hamiltonian structure with other approaches in the literature, such as Clebsch potentials, Lagrangian to Eulerian transformations, and its use in clarifying linearization stability, are discussed.Research supported in part by NSF grant MCS 81-08814(A02)Research supported in part by NSF grant MCS 81-07086     

Classical and quantum mechanical systems of Toda-lattice type

Solutions to the classical periodic and non-periodic Toda lattice type Hamiltonian systems are expressed in terms of an Iwasawa-type factorization of a large Lie group. The scattering of these systems is determined in the non-periodic case. For the generalized periodic Toda lattices a generalization of Kostant's formula is obtained using standard representations of affine Lie groups.Research partially supported by NSF Grant MCS 83-01582Research partially supported by NSF Grant MCS 79-03153     

A general Lee-Yang theorem for one-component and multicomponent ferromagnets

We show that any measure on ⁿ possessing the Lee-Yang property retains that property when multiplied by a ferromagnetic pair interaction. Newman's Lee-Yang theorem for one-component ferromagnets with general single-spin measure is an immediate consequence. We also prove an analogous result for two-component ferromagnets. ForN-component ferromagnets (N 3), we prove a Lee-Yang theorem when the interaction is sufficiently anisotropic.Research supported in part by NSF grant PHY 78-25390 A01Research supported in part by NSF grant PHY 78-23952     

On the probability properties of the density gradient of a randomly moving incompressible medium

We study the statistical properties of the density-field gradient of a passive admixture in a randomly moving medium. A expression for the relative gradient-variation field is derived. It is shown that the moments and practically all realizations of the relative gradient-variation field increase exponentially with time.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 39, No. 5, pp. 597–606, May, 1996.This work was partially supported by Russian Foundation for Fundamental Research grant No. 95-05-14247.