Tosio Kato’s work on non-relativistic quantum mechanics: part 1

@article{Simon2017TosioKW,
  title={Tosio Kato’s work on non-relativistic quantum mechanics: part 1},
  author={Barry Simon},
  journal={Bulletin of Mathematical Sciences},
  year={2017},
  volume={8},
  pages={121-232}
}
  • B. Simon
  • Published 19 October 2017
  • Mathematics, Physics
  • Bulletin of Mathematical Sciences
We review the work of Tosio Kato on the mathematics of non-relativistic quantum mechanics and some of the research that was motivated by this. Topics in this first part include analytic and asymptotic eigenvalue perturbation theory, Temple–Kato inequality, self-adjointness results, and quadratic forms including monotone convergence theorems. 

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References

SHOWING 1-10 OF 707 REFERENCES

Tosio Kato’s Work on Non-relativistic Quantum Mechanics: A Brief Report

  • B. Simon
  • Physics
    Analysis and Operator Theory
  • 2019
Based at a talk given at the Kato Centennial Symposium in Sept. 2017, this article discusses the scientific life and some of the scientific work of T. Kato.

Schrodinger Operators: With Application to Quantum Mechanics and Global Geometry

Self-Adjointness.- Lp-Properties of Eigenfunctions, and All That.- Geometric Methods for Bound States.- Local Commutator Estimates.- Phase Space Analysis of Scattering.- Magnetic Fields.- Electric

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

Power-law corrections to the Kubo formula vanish in quantum Hall systems

In first order perturbation theory conductivity is given by the Kubo formula, which in a Quantum Hall System equals the first Chern class of a vector bundle. We apply the adiabatic theorem to show

Asymptotic completeness for quantum mechanical potential scattering

A new (geometrical) proof is given for the asymptotic completeness of the wave operators and the absence of a singular continuous spectrum of the Hamiltonian for potentials which decrease faster than

A Note on the Adiabatic Theorem Without Gap Condition

We simplify the proof of the adiabatic theorem of quantum mechanics without gap condition of Avron and Elgart by providing an elementary solution of the ‘commutator equation’. In addition, a minor

Classical Action and Quantum N-Body Asymptotic Completeness

The quantum propagation of N-body systems is asymptotically constrained to Lagrangian manifolds corresponding to particular solutions of the free Hamilton-Jacobi equation. This is used to give a

Adiabatic perturbation theory in quantum dynamics

Introduction.- First-order adiabatic theory.- Space-adiabatic perturbation theory.- Applications and extensions.- Quantum dynamics in periodic media.- Adiabatic decoupling without spectral gap.-

On a variant of commutator estimates in spectral theory

Completeness is proved for systems of two or three quantum particles. The proof is based on the following statement on operators in a Hilbert space. If an operator A is bounded with respect to a

Variational methods in relativistic quantum mechanics

This review is devoted to the study of stationary solutions of lin- ear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical
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