Israel Michael Sigal

Return to equilibrium

We study an atom with finitely many energy levels in contact with a heat bath consisting of photons (blackbody radiation) at a temperature T>0. The dynamics of this system is described by a Liouville operator, or thermal Hamiltonian, which is the sum of an atomic Liouville operator, of a Liouville operator describing the dynamics of a free, massless Bose field, and a local operator describing the interactions between the atom and the heat bath. We show that an arbitrary initial state that is normal with respect to the equilibrium state of the uncoupled system at temperature T converges to an equilibrium state of the coupled system at the same temperature, as time tends to +∞ (return to equilibrium).

Solitary Wave Dynamics in an External Potential

We study the behavior of solitary-wave solutions of some generalized nonlinear Schrödinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing inertial motions of stable solitary waves. We consider solutions of the equations with a non-vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that, over a large interval of time, they describe a solitary wave whose center of mass motion is a solution of Newton’s equations of motion for a point particle in the given external potential, up to small corrections corresponding to radiation damping.

The quantum N-body problem

This selective review is written as an introduction to the mathematical theory of the Schrodinger equation for N particles. Characteristic for these systems are the cluster properties of the potential in configuration space, which are expressed in a simple geometric language. The methods developed over the last 40 years to deal with this primary aspect are described by giving full proofs of a number of basic and by now classical results. The central theme is the interplay between the spectral theory of N-body Hamiltonians and the space–time and phase-space analysis of bound states and scattering states.

Geometric methods in the quantum many-body problem. Nonexistence of very negative ions

In this paper we develop the geometric methods in the spectral theory of many-body Schrödinger operators. We give different simplified proofs of many of the basic results of the theory. We prove that there are no very negative ions in Quantum Mechanics.

Asymptotics of the Ground State Energies of Large Coulomb Systems

We prove the Scott conjecture for molecules.

Non-linear Wave and Schrodinger Equations I. Instability of Periodic and Quasiperiodic Solutions*

We investigate stability of periodic and quasiperiodic solutions of linear wave and Schrödinger equations under non-linear perturbations. We show in the case of the wave equations that such solutions are unstable for generic perturbations. For the Schrödinger equations periodic solutions are stable while the quasiperiodic ones are not. We extend these results to periodic solutions of non-linear equations.

Smooth Feshbach map and operator-theoretic renormalization group methods

Abstract A new variant of the isospectral Feshbach map defined on operators in Hilbert space is presented. It is constructed with the help of a smooth partition of unity, instead of projections, and is therefore called smooth Feshbach map . It is an effective tool in spectral and singular perturbation theory. As an illustration of its power, a novel operator-theoretic renormalization group method is described and applied to analyze a general class of Hamiltonians on Fock space. The main advantage of the new renormalization group method over its predecessors is its technical simplicity, which it owes to the use of the smooth Feshbach map.

MINIMAL ESCAPE VELOCITIES

We give a new derivation of the minimal velocity estimates (SiSo1) for unitary evolutions. LetH andA be selfadjoint operators on a Hilbert space H. The starting point is Mourres inequality i(H,A) ≥ � > 0, which is supposed to hold in form sense on the spectral subspace Hof H for some interval � ⊂ R. The second assumption is that the multiple commutators ad (k) A (H) are well- behaved fork = 1...n (n ≥ 2) . Then we show that, for a dense set of s in Hand allm < n−1, t = exp(−iHt) is contained in the spectral subspace A ≥ �t as t → ∞, up to an error of order t −m in norm. We apply this general result to the case where H is a Schrodinger operator on R n and A the dilation generator, proving that t(x) is asymptotically supported in the set |x| ≥ t √ � up to an error of order t −m in norm.

Mathematical theory of nonrelativistic matter and radiation

We consider a system of finitely many nonrelativistic electrons bound in an atom or molecule which are coupled to the electromagnetic field via minimal coupling or the dipole approximation. Among a variety or results, we give sufficient conditions for the existence of a ground state (an eigenvalue at the bottom of the spectrum) and resonances (eigenvalues of a complex dilated Hamiltonian) of such a system. We give a brief outline of the proofs of these statements which will appear at full length in a later work.

Long Time Motion of NLS Solitary Waves in a Confining Potential

Abstract.We study the motion of solitary-wave solutions of a family of focusing generalized nonlinear Schrödinger equations with a confining, slowly varying external potential, V(x).A Lyapunov-Schmidt decomposition of the solution combined with energy estimates allows us to control the motion of the solitary wave over a long, but finite, time interval.We show that the center of mass of the solitary wave follows a trajectory close to that of a Newtonian point particle in the external potential V(x) over a long time interval.Communicated by Rafael D. Benguria