Propagation of singularities for complex-scaled operators on domains with boundary

Establish propagation-of-singularities results in the semiclassical calculus for domains with boundary for operators whose symbols have nonpositive imaginary parts, thereby permitting removal of the assumption that the Cartesian PML scaling function is linear near the truncation boundary.

Background

The analysis of the Cartesian perfectly matched layer requires an a priori estimate near the truncation boundary. For operators with nonpositive imaginary parts in their principal symbols, the necessary propagation-of-singularities theory for semiclassical operators on domains with boundary is not currently available in the literature. The paper therefore assumes that each scaling function is linear near the PML boundary, enabling a reflection argument that extends the solution beyond the boundary and allows the authors to use propagation results for manifolds without boundary.

Developing the missing boundary propagation theory would remove this technical linearity assumption and potentially extend the exponential-accuracy result to more general Cartesian PML scaling functions.

References

For such an operator, the relevant propagation of singularities results in the semiclassical calculus for domains with a boundary have not yet been written down in the literature. Indeed, such propagation of singularities results in the semiclassical calculus have been proved (i) on manifolds without boundary for operators with symbols whose imaginary parts are single-signed Theorem E.47 and (ii) on manifolds with boundary for $P$ . The assumption in \S\ref{s:def} that $F_i'(x)= x$ for $|x|\geq R_{,i}$ allows us to use a reflection argument (similar to that used for the Cartesian PML analyses in 2-d in Proof of Theorem 5.5, Lemma 3.4) to extend the solution past $\partial Q_{}$ and use the propagation results of (i) above, i.e., bypassing the issue of propagation up to the boundary. Once the relevant propagation results are written down in the literature, this assumption can be removed.

— Cartesian PML truncation for the Helmholtz equation is exponentially accurate at high frequency  (2609.11343 - Averseng et al., 10 Sep 2026) in Section 1, subsection “Why we assume that F'_i is linear near the PML boundary”