Barry Simon

Coupling constant analyticity for the anharmonic oscillator

Abstract We study the analytic properties of the singular perturbation theory for p2 + x2 + βx4. We prove rigorously several of the properties of the energy levels, previously found by C. Bender and T. T. Wu using methods of unknown validity. In particular: (a) En(β) has a “global” third-order branch point at β = 0, i.e., any path of continuation which winds three times around β = 0 and circles clockwise about all branch points, returns En to where it started from and a path that makes one turn around does not. (b) On the threesheeted surface, β = 0 is not an isolated singularity; thus, there are infinitely many singularities. (c) The singularities have ± 270° as asymptotic phase. We also show that the perturbation series is asymptotic uniformly in any sector | arg β| with θ 3π 2 . We extend these results to many dimensional oscillators and x2m perturbations. Finally, we study the Pade approximants formed from the Rayleigh-Schrodinger series. Since En(β) has no singularities in the cut plane |argβ|

The Thomas-Fermi theory of atoms, molecules and solids

Abstract We place the Thomas-Fermi model of the quantum theory of atoms, molecules, and solids on a firm mathematical footing. Our results include: (1) A proof of existence and uniqueness of solutions of the nonlinear Thomas-Fermi equations as well as the fact that these solutions minimize the Thomas-Fermi energy functional, (2) a proof that in a suitable large nuclear charge limit, the quantum mechanical energy is asymptotic to the Thomas-Fermi energy, and (3) control of the thermodynamic limit of the Thomas-Fermi theory on a lattice.

Quantum mechanics for Hamiltonians defined as quadratic forms

This monograph combines a thorough introduction to the mathematical foundations of n-body Schrodinger mechanics with numerous new results.Originally published in 1971.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Infrared bounds, phase transitions and continuous symmetry breaking

We present a new method for rigorously proving the existence of phase transitions. In particular, we prove that phase transitions occur in (φ·φ)32 quantum field theories and classical, isotropic Heisenberg models in 3 or more dimensions. The central element of the proof is that for fixed ferromagnetic nearest neighbor coupling, the absolutely continuous part of the two point function ink space is bounded by 0(k−2). When applicable, our results can be fairly accurate numerically. For example, our lower bounds on the critical temperature in the three dimensional Ising (resp. classical Heisenberg) model agrees with that obtained by high temperature expansions to within 14% (resp. a factor of 9%).

Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians

Abstract Abstract connections between integral kernels of positivity preserving semigroups and suitable Lp contractivity properties are established. Then these questions are studied for the semigroups generated by −Δ + V and H Ω , the Dirichlet Laplacian for an open, connected region Ω. As an application under a suitable hypothesis, Sobolev estimates are proved valid up to ∂Ω, of the form ¦η(x)¦⩽ cϑ 0 (x) ∥H Ω k η∥ 2 , where ϑ0 is the unique positive L2 eigenfunction of H Ω .

The bound state of weakly coupled Schrodinger operators in one and two dimensions

We study the unique bound state which (-d/sup 2//dx/sup 2/)+lambdaV and -..delta..+lambdaV (in two dimensions) have when lambda is small and V is suitable. Our main results give necessary and sufficient conditions for there to be a bound state when lambda is small and we prove analyticity (resp. nonanalyticity) of the energy eigenvalue at lambda=0 in one (resp. two) dimensions. (AIP)

The Hartree-Fock Theory for Coulomb Systems

For neutral atoms and molecules and positive ions and radicals, we prove the existence of solutions of the Hartree-Fock equations which minimize the Hartree-Fock energy. We establish some properties of the solutions including exponential falloff.

Schrödinger operators with magnetic fields. III. Atoms in homogeneous magnetic field

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 hasLz = 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 largeLz.

Quadratic form techniques and the Balslev-Combes theorem

We extend the theorem of Balslev and Combes on the absence of singular continuous spectrum to a class of interactions includingr−α(3/2≦α<2) local potentials. In addition, we note that the theory of sectorial operators allows a simplification of their proof and allows one to push the cuts through angles larger than the π/2 restriction employed by Balslev-Combes.

The definition of molecular resonance curves by the method of exterior complex scaling

Abstract We introduce an extension of complex scaling which is applicable to molecules in the Born-Oppenheimer approximation and which reduces to the usual complex scaling when that theory is applicable.