Tosio Kato's Work on Non--Relativistic Quantum Mechanics

2017/11/01 by Barry Simon, Simon, Barry
Mathematics · Physics and Astronomy · #35Q40 #46N50 #47A55 #81Q10 #81Q15 #81U05 #Advanced Operator Algebra Research #Classical mechanics #FOS: Mathematics #FOS: Physical sciences #History and Overview (math.HO) #Mathematical Physics (math-ph) #Physics #Quantum #Quantum Mechanics and Applications #Quantum dissipation #Quantum mechanics #Relativistic mechanics #Relativistic quantum mechanics #Spectral Theory in Mathematical Physics #Theoretical physics #Theory of relativity #Work (physics) #math-ph #math.HO #math.MP #msc:35Q40 #msc:46N50 #msc:47A55 #msc:81Q10 #msc:81Q15 #msc:81U05

paper · pdf · doi:10.48550/arxiv.1711.00528

This is intended for Bull. Math. Sci. which intends to split it in two parts

arxiv created 2017/11/01 · openalex publication_date 2017/11/01 · arxiv updated 2017/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 include analytic and asymptotic eigenvalue perturbation theory, Temple--Kato inequality, self--adjointness results, quadratic forms including monotone convergence theorems, absence of embedded eigenvalues, trace class scattering, Kato smoothness, the quantum adiabatic theorem and Kato's ultimate Trotter Product Formula.

Citations

• Mémoire sur les séries divergentes
• Spectral properties of Schrödinger operators and time-decay of the wave functions
• On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure
• Variational Principles for Scattering Processes. I
• On the product of semi-groups of operators
• Perturbation series for eigenvalues of Analytic Non-symmetric Operators
• Quasi-adiabatic Continuation of Quantum States: The Stability of Topological Ground State Degeneracy and Emergent Gauge Invariance
• Zero energy asymptotics of the resolvent for a class of slowly decaying potentials
• Essential self-adjointness of powers of generators of hyperbolic equations
• Spectral Analysis of N-Body Schrodinger Operators
• Quantized Hall Conductance in a Two-Dimensional Periodic Potential
• Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren
• Decay estimates for Schrödinger operators
• Note on product formulas for operator semigroups
• A class of analytic perturbations for one-body Schrödinger Hamiltonians
• Quantal phase factors accompanying adiabatic changes
• Supersymmetry and Morse theory
• Mean-field limit of Bose systems: rigorous results
• Resonances in n-Body Quantum Systems With Dilatation Analytic Potentials and the Foundations of Time-Dependent Perturbation Theory
• Hardy inequalities, Rellich inequalities and local Dirichlet forms

Related

• The adiabatic theorem for non-Hermitian quantum systems with real eigenvalues and the complex geometric phase
• Recent developments in non-perturbative quantum field theory
• A mathematical perspective on the phenomenology of non-perturbative Quantum Field Theory
• A Course on: "An Algebraic Approach to Nonperturbative Quantum Field Theory"
• Non-Archimedean Mathematics and the formalism of Quantum Mechanics