Mourre theory for analytically fibered operators revisited

TL;DR

This paper revisits and refines Mourre theory for analytically fibered operators, addressing previous limitations in the construction of conjugate operators and ensuring the theory's applicability.

Contribution

It provides a modified construction of conjugate operators that overcomes earlier regularity issues, enhancing Mourre theory's applicability to fibered Hamiltonians.

Findings

Corrected the regularity assumptions needed for Mourre theory

Extended the applicability of the conjugate operator construction

Clarified the conditions under which Mourre theory holds for fibered operators

Abstract

About 25 years ago our article "Mourre theory for analytically fibered operators" was published in J. of Functional Analysis. This article proposed a general construction of a conjugate operator for a wide class of self-adjoint analytically fibered hamiltonians, provided that one accepts a more accurate notion of threshold. It is only recently that Olivier Poisson mentionned us a problem with the statement that H 0 $\in$ C $\infty$ (A I). Actually even H 0 $\in$ C 2 (A I) or H 0 $\in$ C 1+0 (A I) , which is crucial for the full application of Mourre theory, is problematic with our initial construction. However the statement and the construction can be modified in order to make work all the theory. This is explained here.