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 h -dependent delta-function potentials on R \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 Im ⁡ z ∼ − γ h log ⁡ ( 1 / h ) . \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 γ . \gamma . We present numerical evidence of the existence of more and more possible values of γ \gamma for larger numbers of delta poles.