Spectral and resonance properties of Smilansky Hamiltonian

TL;DR

This paper investigates the spectral and resonance characteristics of the Smilansky Hamiltonian, revealing new insights into its discrete spectrum and resonance structure through asymptotic analysis and numerical methods.

Contribution

It provides the first weak-coupling asymptotics for the ground state and uncovers a complex resonance structure in the model.

Findings

Derived weak-coupling asymptotics for the ground state

Numerically identified the discrete spectrum

Revealed a rich resonance structure

Abstract

We analyze the Hamiltonian proposed by Smilansky to describe irreversible dynamics in quantum graphs and studied further by Solomyak and others. We derive a weak-coupling asymptotics of the ground state and add new insights by finding the discrete spectrum numerically. Furthermore, we show that the model has a rich resonance structure.