Horng-Tzer Yau

The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics

Starting with a “relativistic” Schrödinger Hamiltonian for neutral gravitating particles, we prove that as the particle numberN→∞ and the gravitation constantG→0 we obtain the well known semiclassical theory for the ground state of stars. For fermions, the correct limit is to fixGN2/3 and the Chandrasekhar formula is obtained. For bosons the correct limit is to fixGN and a Hartree type equation is obtained. In the fermion case we also prove that the semiclassical equation has a unique solution — a fact which had not been established previously.

Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics

We prove that the spectral gap of the Kawasaki dynamics shrink at the rate of 1/L2 for cubes of sizeL provided that some mixing conditions are satisfied. We also prove that the logarithmic Sobolev inequality for the Glauber dynamics in standard cubes holds uniformly in the size of the cube if the Dobrushin-Shlosman mixing condition holds for standard cubes.

Stability of Coulomb systems with magnetic fields. III: Zero energy bound states of the Pauli operator

It is shown that there exist magnetic fields of finite self energy for which the operator σ·(p−A) has a zero energy bound state. This has the consequence that single electron atoms, as treated recently by Fröhlich, Lieb, and Loss [1], collapse when the nuclear charge numberz≧9π2/8α2 (α is the fine structure constant).

Relaxation of excited states in nonlinear Schrödinger equations

We consider a nonlinear Schrodinger equation in

STABLE DIRECTIONS FOR EXCITED STATES OF NONLINEAR SCHRÖDINGER EQUATIONS

\R^3

The

with a bounded local potential. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data is small and is near some nonlinear {\it excited} state. We give a sufficient condition on the initial data so that the solution to the nonlinear Schrodinger equation approaches to certain nonlinear {\it ground} state as the time tends to infinity.

N^{7/5}

ABSTRACT We consider nonlinear Schrödinger equations in . Assume that the linear Hamiltonians have two bound states. For certain finite codimension subset in the space of initial data, we construct solutions converging to the excited states in both non-resonant and resonant cases. In the resonant case, the linearized operators around the excited states are non-self adjoint perturbations to some linear Hamiltonians with embedded eigenvalues. Although self-adjoint perturbation turns embedded eigenvalues into resonances, this class of non-self adjoint perturbations turn an embedded eigenvalue into two eigenvalues with the distance to the continuous spectrum given to the leading order by the Fermi golden rule.

law for charged bosons

Non-relativistic bosons interacting with Coulomb forces are unstable, as Dyson showed 20 years ago, in the sense that the ground state energy satisfiesE0≦−AN7/5. We prove that 7/5 is the correct power by proving thatE0≧−BN7/5. For the non-relativistic bosonic, one-component jellium problem, Foldy and Girardeau showed thatE0≦−CNρ1/4. This 1/4 law is also validated here by showing thatE0≧−DNρ1/4. These bounds prove that the Bogoliubov type paired wave function correctly predicts the order of magnitude of the energy.

Logarithmic Sobolev inequality for lattice gases with mixing conditions

AbstractLet