Newton polygons and resonances of multiple delta-potentials

Abstract:

We prove explicit asymptotics for the location of semiclassical scattering resonances in the setting of $h$-dependent delta-function potentials on $\mathbb {R}$. In the cases of two or three delta poles, we are able to show that resonances occur along specific lines of the form $\operatorname {Im}z \sim -\gamma h \log (1/h).$ More generally, we use the method of Newton polygons to show that resonances near the real axis may only occur along a finite collection of such lines, and we bound the possible number of values of the parameter $\gamma .$ We present numerical evidence of the existence of more and more possible values of $\gamma$ for larger numbers of delta poles.