Explicit Semiclassical Resonances from Many Delta Functions

We study the scattering resonances arising from multiple $h$-dependent Dirac delta functions on the real line in the semiclassical regime $h \rightarrow 0$. We focus on resonances lying in strings along curves of the form $\text{Im } z \sim -\gamma h\log(1/h)$ and find that resonances along such strings exist if and only if $\gamma$ is a slope of a Newton polygon we construct from the parameters. Furthermore, the set of these $\gamma$ corresponds to a complete and disjoint partitioning of a line segment with delta functions at interval endpoints. Hence, there are at most $N-1$ strings of resonances from $N$ delta functions, improving a bound from (Datchev, Marzuola, & Wunsch 2023). Lastly, we identify a `dominant pair' of delta functions in the sense that they correspond to the longest-living string of resonances, this string is the only one of logarithmic shape with respect to $\text{Re } z$, and no delta functions between them can contribute to strings of resonances. The simple properties of delta functions permit elementary proofs, and we provide many visual examples to demonstrate the results.