# 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} }

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.

## 21 Citations

### Differential equations of quantum mechanics

- Physics, MathematicsQuarterly of Applied Mathematics
- 2022

We review very briefly the main mathematical structures and results in some important areas of Quantum Mechanics involving PDEs and formulate open problems.

### The Feshbach–Schur map and perturbation theory

- MathematicsPartial Differential Equations, Spectral Theory, and Mathematical Physics
- 2021

This paper deals with perturbation theory for discrete spectra of linear operators. To simplify exposition we consider here self-adjoint operators. This theory is based on the Feshbach-Schur map and…

### Representation of non-semibounded quadratic forms and orthogonal additivity

- MathematicsJournal of Mathematical Analysis and Applications
- 2018

### Self-adjointness of non-semibounded covariant Schr\"odinger operators on Riemannian manifolds

- Mathematics
- 2021

In the context of a geodesically complete Riemannian manifold M , we study the self-adjointness of ∇∇+V where ∇ is a metric covariant derivative (with formal adjoint ∇) on a Hermitian vector bundle V…

### Twelve tales in mathematical physics: An expanded Heineman prize lecture

- PhysicsJournal of Mathematical Physics
- 2022

This is an extended version of my 2018 Heineman prize lecture describing the work for which I got the prize. The citation is very broad, so this describes virtually all my work prior to 1995 and some…

### Spectral triples, Coulhon-Varopoulos dimension and heat kernel estimates

- Mathematics
- 2022

We connect the (completely bounded) local Coulhon-Varopoulos dimension to the spectral dimension of spectral triples associated to sub-Markovian semigroups (or Dirichlet forms) acting on classical…

### Quantum theory and functional analysis

- Physics
- 2019

Quantum theory and functional analysis were created and put into essentially their final form during similar periods ending around 1930. Each was also a key outcome of the major revolutions that both…

### Sharp bound for embedded eigenvalues of Dirac operators with decaying potentials

- Mathematics
- 2022

. We study eigenvalues of the Dirac operator with canonical form where 𝑝 and 𝑞 are real functions. Under the assumption that the essential spectrum of 𝐿 𝑝,𝑞 is (−∞,∞) . We prove that 𝐿 𝑝,𝑞…

### Extensions of symmetric operators that are invariant under scaling and applications to indicial operators

- Mathematics
- 2021

Indicial operators are model operators associated to an elliptic di erential operator near a corner singularity on a strati ed manifold. These model operators are de ned on generalized tangent cone…

### UNBOUNDED EIGENVALUE PROBLEMS IN THE GENERALIZED FORM

- Mathematics
- 2019

Unbounded eigenvalue problems in the generalized form are reformulated, implicitly leading to bounded standard eigenvalue problems. By performing implicit linear fractional transformations, the way…

## References

SHOWING 1-10 OF 707 REFERENCES

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

- PhysicsAnalysis 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

- Mathematics
- 1987

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

- Mathematics
- 1982

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

- Physics, Mathematics
- 1990

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

- Mathematics
- 1978

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

- Physics, Mathematics
- 2001

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

- Mathematics
- 1997

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

- Physics, Mathematics
- 2003

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

- Mathematics
- 1990

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

- Physics, Mathematics
- 2007

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…