Explicit Semiclassical Resonances from Many Delta Functions

TL;DR

This paper characterizes the distribution of semiclassical scattering resonances from multiple delta functions on the real line, revealing a geometric structure linked to Newton polygons and improving bounds on the number of resonance strings.

Contribution

It introduces a geometric framework using Newton polygons to precisely determine the existence and structure of resonance strings from multiple delta functions, and identifies the dominant pair responsible for the longest-living resonances.

Findings

Resonances form strings along curves with specific imaginary parts related to Newton polygon slopes.

Number of resonance strings is at most one less than the number of delta functions, improving previous bounds.

A dominant pair of delta functions produces the longest-living resonance string, unique in its logarithmic shape.

Abstract

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.