**Abbreviations**

The following abbreviations are used in this manuscript:


### **Appendix A. Scattering Theory, Phase-Shifts and the Friedel Sum Rule**

In this Appendix we review some useful results from scattering theory. In Appendix A.1, we provide the definition of the S- and T-matrix in scattering theory. In Appendix A.2, we derive the Friedel sum rule for a non-interacting electron gas with an elastic impurity. In Appendix A.3, we illustrate these concepts on the simple case of a chiral edge state tunnel coupled to a single resonant level. In Appendix A.4, we derive the phase-shift induced on lead electrons by the scattering potential of the LFL, Equation (8).
