William G. Faris

Inequalities and uncertainty principles

Sobolev inequalities give lower bounds for quantum mechanical Hamiltonians. These inequalities are derived from commutator inequalities related to the Heisenberg uncertainty principle.

Commutators and self-adjointness of Hamiltonian operators

A time dependent approach to self-adjointness is presented and it is applied to quantum mechanical Hamiltonians which are not semi-bounded. Sufficient conditions are given for self-adjointness of Schrödinger and Dirac Hamiltonians with potentials which are unbounded at infinity. The method is the introduction of an auxiliary operatorN≧0 whose rate of change (commutator with the Hamiltonian) is bounded by a multiple ofN.

Invariant Cones and Uniqueness of the Ground State for Fermion Systems

A perturbation theory is developed for self‐adjoint operators whose resolvents leave a cone invariant. The Perron‐Frobenius theory may be applied to the perturbed operator to conclude that its lowest eigenvalue has multiplicity one. In quantum field theory this gives uniqueness of the vacuum for a class of fermion interactions different from that considered by Gross.

Degenerate and non-degenerate ground states for Schrödinger operators

In the study of quantum mechanical energy operators, particular attention is paid to the question of whether the bottom of the spectrum is an eigenvalue and whether that eigenvalue is simple. The corresponding eigenvector is usually called the gound state. Throughout our discussion we will say that a self-adjoint operator H has no ground state degeneracy when either E inf spectrum H is not an eigenvalue or E is a simple eigenvalue. Our concern in this note will be SchrSdinger operators H -A V acting in where V is a suitable multiplication operator. When n 1 the lack of ground state degeneracy for a large class of V is classical theorem in the theory of ordinary differential equations. There is n rgument in Courant and Hilbert [4] concerning the case of general n. Implicit in their rguments are assumptions (about the regularity of nodes of any eigenvector) that seem difficult to prove for general V. (However, we should remark that a proof along the lines of Courant and Hilbert should be possible, especially if one uses Kato’s inequality [14].) The modern treatment of the non-degeneracy problem follows an idea of Glimm and Jaffe [11] in their study of quantum field models. They suggested applying theorems of PerronFrobenius [10, 17] type to exp(-tH). The applicability of these ideas to SchrSdinger operators was noted by Simon and HSegh-Krohn [23] and Faris (see [2]). It was discussed further in Faris [8], where the following general result appears.

Product formulas for perturbations of linear propagators

The abstract equation of evolution for a Banach-space-valued function u(t) may be written du(t)/dt = c(t) u(t) where for each t, C(t) is a linear operator acting in the Banach space. The abstract Cauchy problem is to establish existence and uniqueness of solutions with prescribed initial data U(S) = Y, which depend continuously on the data. If C(t) = C is independent of the time t, the solution should be u(t) = exp ((t - s) C) Y. The Hille-Yosida theorem of semigroup theory may be used to give sense to this exponential. In general the solution is not given by such a simple formal expression. There should exist, however, a family of continuous linear operators R(t, s) such that u(t) = R(t, s) Y is the solution with initial condition U(S) = Y. Such a family is called a propagator or a Green’s operator. Conditions on C(t) which guarantee the existence of the family R(t, s) were first given by Kato [6]. The problem has been further studied by Yosida [Z5; Chap. XIV], Nelson [II], Kisyliski [7], and Lions [S]. Now assume that C(t) = A(t) + B(t). The problem of perturbation theory considered here is to express R(t, s) explicitly in terms of A(t) and B(t) and solutions of equations involving these operators separa- tely. Such general formulas were obtained by Phillips [12] and Segal [Z3]. Th eir method was essentially iteration of the integral equation

Diffusion, Quantum Theory, and Radically Elementary Mathematics

Diffusive motion--displacement due to the cumulative effect of irregular fluctuations--has been a fundamental concept in mathematics and physics since Einsteins work on Brownian motion. It is also relevant to understanding various aspects of quantum theory. This book explains diffusive motion and its relation to both nonrelativistic quantum theory and quantum field theory. It shows how diffusive motion concepts lead to a radical reexamination of the structure of mathematical analysis. The books inspiration is Princeton University mathematics professor Edward Nelsons influential work in probability, functional analysis, nonstandard analysis, stochastic mechanics, and logic. The book can be used as a tutorial or reference, or read for pleasure by anyone interested in the role of mathematics in science. Because of the application of diffusive motion to quantum theory, it will interest physicists as well as mathematicians. The introductory chapter describes the interrelationships between the various themes, many of which were first brought to light by Edward Nelson. In his writing and conversation, Nelson has always emphasized and relished the human aspect of mathematical endeavor. In his intellectual world, there is no sharp boundary between the mathematical, the cultural, and the spiritual. It is fitting that the final chapter provides a mathematical perspective on musical theory, one that reveals an unexpected connection with some of the books main themes.

The stochastic Heisenberg model

Abstract The stochastic Heisenberg model is a classical mechanical model for a magnet in contact with a medium at fixed temperature. The dynamics in the infinite volume limit is a well-defined Markov process. Equilibrium states exist, and in this limit there is the possibility of multiple equilibrium states at low temperature, corresponding to different directions of magnetism.

Spin correlation in stochastic mechanics

Stochastic mechanics may be used to described the spin of atomic particles. The spin variables have the same expectations as in quantum mechanics, but not the same distributions. They play the role of hidden variables that influence, but do not determine, the results of Stern-Gerlach experiments involving magnets. During the course of such an experiment spin becomes correlated with position. The case of two particles with zero total spin occurs in Bohms version of the Einstein-Rosen-Podolsky experiment.

A Quantum Crystal with Multidimensional Anharmonic Oscillators

The quantum anharmonic crystal is made up of a large number of multidimensional anharmonic oscillators arranged in a periodic spatial lattice with a nearest neighbor coupling. If the coupling coefficient is sufficiently small, then there is a convergent expansion for the ground state of the crystal. The estimates on the convergence are independent of the size of the crystal. The proof uses the path integral representation of the ground state in terms of diffusion processes. The convergence of the cluster expansion depends on the ergodicity properties of these processes.

Essential self-adjointness of operators in ordered Hilbert space

LetH0≧0 be a self-adjoint operator acting in a spaceL2(M, μ). It is assumed thatH0e=0, wheree is strictly positive, and that exp(−tH0) is positivity preserving fort≧0. LetV be a real function onM such that its positive part is inL2(M,e2μ) and its negative part is relatively small with respect toH0. ThenH=H0+V is essentially self-adjoint on the intersection of the domains ofH0 andV. This result is applied to Schrödinger operators and to quantum field Hamiltonians.