Infinite-dimensional analyticity in quantum physics
- URL: http://arxiv.org/abs/2108.10094v1
- Date: Mon, 23 Aug 2021 11:49:10 GMT
- Title: Infinite-dimensional analyticity in quantum physics
- Authors: Paul E. Lammert
- Abstract summary: A study is made, of families of Hamiltonians parameterized over open subsets of Banach spaces.
It renders many interesting properties of eigenstates and thermal states analytic functions of the parameter.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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.
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