Computing resonances based on the pole condition


open resonators

slat noise

resonances in wave-guides

quantum resonances
 

Part 1: Theory of resonances


basic notions

definition of resonances

illustration

time-dependent problem

stability

Aguilar-Balslev-Combes-Simon Theory
 

Part 2: Radiation conditions


radiation conditions for bounded obstacles

rough surfaces

wave guides

transparent boundary conditions

radiation conditions for complex wave numbers
 

Part 3: Pole condition


references on pole condition

Laplace transform of Helmholz equation

pole condition

solvability of Volterra equation

representation formula

equivalence to Sommerfeld's radiation condition

pole condition for rough surface scattering
 

Part 4: Perfecly Match Layer (PML) method


PML: radiation condition

PML: TBC

relation to PML
 

Part 5: Transparent boundary conditions based on the pole condition


Volterra equation

solution of boundary value problem

solution of coupled problem

real axis method

pole condition for complex k

discretization

matrix structure

numerical convergence results
 

Part 6: Numerical experiments with resonances

a one-dimensional problem

spectra for PML and pole condition

resonances in a 2D open system

ellipse in wave-guide

discrete spectrum