Infinite-dimensional analyticity in quantum physics

TL;DR

This paper develops a framework for analyzing families of quantum Hamiltonians parameterized over infinite-dimensional Banach spaces, establishing conditions for their eigenstates and thermal states to vary analytically with parameters.

Contribution

It generalizes Kato's theory to infinite-dimensional settings, providing criteria for holomorphy of Hamiltonian families and their associated resolvent and exponential maps.

Findings

Established conditions for sectorial families of Hamiltonians to be analytic

Proved analyticity of resolvent and exponential maps in parameterized Hamiltonians

Applied theory to quantum systems with scalar/vector potentials and interactions

Abstract

A study is made, of families of Hamiltonians parameterized over open subsets of Banach spaces in a way which renders many interesting properties of eigenstates and thermal states analytic functions of the parameter. Examples of such properties are charge/current densities. The apparatus can be considered a generalization of Kato's theory of analytic families of type B insofar as the parameterizing spaces are infinite dimensional. It is based on the general theory of holomorphy in Banach spaces and an identification of suitable classes of sesquilinear forms with operator spaces associated with Hilbert riggings. The conditions of lower-boundedness and reality appropriate to proper Hamiltonians is thus relaxed to sectoriality, so that holomorphy can be used. Convenient criteria are given to show that a parameterization $x \mapsto {\mathsf{h}}_x$ of sesquilinear forms is of the required sort ({\it regular sectorial families}). The key maps ${\mathcal R}(\zeta,x) = (\zeta - H_x)^{-1}$ and ${\mathcal E}(\beta,x) = e^{-\beta H_x}$, where $H_x$ is the closed sectorial operator associated to ${\mathsf {h}}_x$, are shown to be analytic. These mediate analyticity of the variety of state properties mentioned above. A detailed study is made of nonrelativistic quantum mechanical Hamiltonians parameterized by scalar- and vector-potential fields and two-body interactions.