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 in this first part include analytic and asymptotic eigenvalue perturbation theory, Temple–Kato inequality, self-adjointness results, and quadratic forms including monotone convergence theorems.
Highlights
Note: There are four pictures in this part and one picture in Part 2
In 2017, we are celebrating the 100th anniversary of the birth of Tosio Kato
While there can be arguments as to which of his work is the deepest or most beautiful, there is no question that the most significant is his discovery, published in 1951, of the self-adjointness of the quantum mechanical Hamiltonian for atoms and molecules [314]. This is the founding document and Kato is the founding father of what has come to be called the theory of Schrödinger operators
Summary
Note: There are four pictures in this part and one picture in Part 2. Kato [345, Section VII.1.2] has a more general definition that applies even to closed operators between two Banach spaces X and Y but he proves that it is equivalent to the above definition so long as X = Y and every A(β) has a non-empty resolvent set (which is no restriction if you want to consider isolated eigenvalues). With this definition, all the eigenvalue perturbation theory for the bounded case carries over since λ0 is a discrete eigenvalue of A(β0) if and only if (λ0 − z0)−1 is a discrete eigenvalue of ( A(β0) − z0)−1. The set of algebraic terms obtained by the above proof are the same for asymptotic and analytic perturbation theory so the an are given by Rayleigh–Schrödinger perturbation theory
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