MATHEMATICAL PHYSICS


Faculty:

Using the tools and standards of rigor of contemporary mathematics, mathematical physicists study problems of modern theoretical physics. Some compensation for the fact that mathematicians tend to call them physicists and that physicists tend to call them mathematicians is provided by the breadth of physical subject matter and beauty of various unexpected interconnections in the mathematical structure of rather distinct physical situations. Among the areas of contemporary mathematics of greatest relevance to these studies are functional analysis and probability theory. The field has concentrated in three main areas: (1) quantum field theory, especially the precise construction of models; (2) statistical mechanics, especially the theory of phase transitions; and (3) nonrelativistic quantum mechanics (Schrödinger operators), including the connections to atomic and molecular physics. Barry Simon has worked in all three areas, although his current interests lie primarily in (3) and to a lesser extent in (2).

Current activity at Caltech in the mathematical side of nonrelativistic quantum mechanics has primarily involved two areas. One is the study of inverse spectral problems using a new approach based on a Krein formula. The other is the study of Schrödinger operators with random and almost periodic potentials, where some rather subtle mathematics is being uncovered. There are a number of potential applications of such models to various condensed matter situations.

Selected Recent Publications

  1. Operators with singular continuous spectrum: I. General operators, Ann. Math. 141 (1995), 131-145
  2. Representations of Finite and Compact Groups, Graduate Studies in Mathematics 10, American Mathematical Society, 1996
  3. (with A. Gordon, S. Jitomirskaya and Y. Last) Duality and singular continuous spectrum in the almost Mathieu equation, Acta Math. 178 (1997), 169-183
  4. (with Y. Last) Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999), 329-367
  5. A new approach to inverse spectral theory, I. Fundamental formalism, Annals of Math. 150 (1999), 1029-1057
  6. (with F. Gesztesy) A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure, Annals of Math. 152 (2000), 593-643
  7. (with R. Killip) Sum rules for Jacobi matrices and their applications to spectral theory, preprint

Selected Older Publications

  1. (with M. Reed) Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, Academic Press, 1972; Vol. II: Fourier Analysis, Self-Adjointness, Academic Press, 1975; Vol. III: Scattering Theory, Academic Press, 1978; Vol. IV: Analysis of Operators, Academic Press, 1977
  2. (with F. Guerra and L. Rosen) The P(\phi)_2 quantum theory as classical statistical mechanics, Ann. Math. 101 (1975), 111-259
  3. (with E. Lieb) The Thomas-Fermi theory of atoms, molecules and solids, Adv. Math. 23 (1977), 22-116
  4. (with J. Fröhlich and T. Spencer) Infrared bounds, phase transitions and continuous symmetry breaking, Commun. Math. Phys. 50 (1976), 79-85
  5. (with P. Perry and I. Sigal) Spectral analysis of multiparticle Schrödinger operators, Ann. Math. 114 (1981), 519-567
  6. (with M. Aizenman) Brownian motion and Harnack's inequality for Schrödinger operators, Commun. Pure Appl. Math. 35 (1982), 209-273
  7. Semiclassical analysis of low lying eigenvalues, II. Tunneling, Ann. Math. 120 (1984), 89-118
  8. Holonomy, the quantum adiabatic theorem and Berry's phase, Phys. Rev. Lett. 51 (1983), 2167-2170
  9. (with T. Wolff) Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Commun. Pure Appl. Math. 39 (1986), 75-90

Barry Simon's

Contents contains links to the other Physics departments.

More information may be found at the following WWW addresses:
PMA Home Page: http://www.pma.caltech.edu
Caltech Home Page: http://www.caltech.edu