Validity of the Dyson expansion under operator-bounded perturbations

Establish whether the formal Dyson series representation for the semigroup e^{-t(A+B)}—namely, e^{-t(A+B)} equals e^{-tA} plus the series over n≥1 of integrals over the simplex Δ_n^t of the product e^{-t_0 A} B e^{-t_1 A} ⋯ B e^{-t_n A}—holds for all t>0 when A is a non-negative self-adjoint operator and B is hermitian and operator-bounded relative to A; in particular, determine if this series converges and defines e^{-t(A+B)} as an operator identity in full generality.

Background

The paper seeks to justify a Dyson-type expansion for semigroups generated by A+B when B is only relatively form-bounded, a setting in which the traditional Dyson series is not straightforwardly justified. The authors construct an alternative convergent expansion via contour integration and Neumann series that avoids directly invoking the classical Dyson series term-by-term.

Crucially, they remark that even under the stronger hypothesis that B is operator-bounded with respect to A, the general validity of the classical Dyson expansion formula is unknown. They cite prior work that provides partial results, indicating that a full resolution of the Dyson expansion’s validity under operator-bounded perturbations remains an open problem. Resolving this would clarify when the widely used formal series yields a correct operator identity for e{-t(A+B)}.

References

Note that even under the stronger assumption that $B$ is operator-bounded relative to $A$, the validity of abstract:dyson is not known, in general; we refer to for partial results in this direction.

abstract:dyson:

e−t(A+B)=e−tA+∑n≥1(−1)n∫Δnte−t0ABe−t1A⋯Be−tnAdt‾e^{-t(A+B)} = e^{-t A} + \sum_{n\geq 1}(-1)^n \int_{\Delta_n^t} e^{-t_0 A} B e^{-t_1 A} \cdots B e^{-t_n A} d\underline t

— Dyson expansion for form-bounded perturbations, and applications to the polaron problem (2512.13443 - Desio et al., 15 Dec 2025) in Section 2 (Abstract Dyson Expansion for Form-Bounded Perturbations), paragraph following Equation (2.1)