# Resonant-State Expansion of the Fano Peak in Open Quantum Systems

## TL;DR

This paper develops a resonant-state expansion method to analytically describe Fano peaks in open quantum systems, revealing how different resonant interactions produce characteristic asymmetries in conductance.

## Contribution

It introduces a novel complete set of states including resonant states, enabling an analytical expansion of conductance without background integrals, and classifies Fano asymmetries based on resonant state interactions.

## Key findings

- Resonant states dominate electron transmission in quantum dots.
- Three types of Fano asymmetry are identified and characterized.
- Microscopic expressions for Fano parameters are derived.

## Abstract

We describe the Fano asymmetry by expanding the transmission amplitude with respect to states with point spectra (discrete eigenstates), including not only bound states but also resonant states with complex eigenvalues. We first introduce a novel complete set that spans the Hilbert space of the central part of an open quantum-dot system. This complete set contains all states of point spectra, but does not contain any states of continuous spectra. We thereby analytically expand the conductance of the dot in terms of all discrete states without any background integrals. This expansion implies that the resonant states produce the main contributions to the electron transmission. We then explain the Fano peak as an interference effect involving resonant states. We find that there are three types of Fano asymmetry according to their origins: the interference between a resonant state and an anti-resonant state, that between a resonant state and a bound state, and that between two resonant states. We derive microscopic expressions of the Fano parameters that describe the three types of Fano asymmetry. We show that the last two types display the asymmetric energy dependence given by Fano, but the first one shows a slightly different form.