Boundary Element Method for Quantum Billiards

Spectrum solver for 2D Helmholtz quantum billiard problems with Dirichlet boundary conditions. Implements a Nyström-based Boundary Element Method (BEM) to construct boundary data, assemble boundary integral operators, and scan wavenumbers to approximate resonant modes. Includes wavefunction visualisation and spectral analysis tools.

This is a refined version of code originally developed for my master's project (text found in rsc/docs/). It's still a good companion to this repo as the methods are identical, and it provides a comprehensive introduction to the theory involved.

Overview

Quantum billiards describe particles confined in a 2D region with perfectly reflecting walls. The dynamics of these systems are described by the Helmholtz equation under Dirichlet boundary conditions

$$(\nabla^2 + k^2)\psi = 0, \quad \psi|_{\partial \Omega} = 0.$$

This equation gives rise to resonant eigenfrequencies $k_i$ which correspond to the energy spectrum of the system ($E_i=k_i^2$). This project applies a Boundary Element Method (BEM) to convert this equation into the Boundary Integral Equation

$$\frac{1}{2}\frac{\partial \psi}{\partial \mathbf{n}}(r_{x}) = \int_{\partial\Omega} \frac{\partial G(r_{x},r_{y})}{\partial \mathbf{n}_{x}} \frac{\partial \psi}{\partial \mathbf{n}}(r_{y}) \; \mathrm{d}s_{y},$$

where

$$G_{k}(\mathbf{r}, \mathbf{r'}) = \frac{i}{4} H_{0}^{(1)}(k|\mathbf{r}- \mathbf{r'}|)$$

is the 2D free-space Green’s function. This equation is approximated to a matrix equation of the form $A\phi = 0$, where $\phi = \frac{\partial \psi}{\partial r}$, using the Nyström quadrature method. Solutions are deduced by scanning across a range of $k$ and detecting singularities in the boundary operator $A$ via SVD minima. Evaluating at individual resonant modes and solving for $\frac{\partial \psi}{\partial r}$ along the boundary allows us to reconstruct physically relevant wavefunction solutions $\psi$ on the billiard as seen below. More generally, evaluating the spacing (Wigner-Dyson) statistics of the spectra of billiard systems allows us to draw parallels between classical and quantum notions of chaos.

Features

• ✅ Boundary element discretisation of the 2D Helmholtz equation
• ✅ Multithreaded BEM matrix assembly
• ✅ Optimised Kernel function via Hankel function tabulation
• ✅ Iterative hot-loop Krylov method for fast resonance scanning via singular value analysis
• ✅ Visualisation of wavefunctions inside the billiard
• ✅ Spectrum unfolding using Weyl’s law
• ✅ Spacing distribution and spectral statistics analysis
• ✅ Support for circle, square, rectangle, cardioid, and Bunimovitch stadium geometries

Directory Structure

• BEM/
  • Mesh/ -- boundary and mesh data construction
  • Kernel.jl -- kernel function
  • Matrix.jl -- matrix operator constuction
  • Resonances.jl -- resonant mode detection
  • Solver. jl -- boundary data and wavefunction solver
  • Optimisations/
    • Hankel.jl -- Hankel function tabulation
    • SingularValues.jl -- Krylov iteration-based min svg finder
• Spectral/
  • Weyl.jl -- spectrum unfolding and counting function
  • WignerDyson.jl -- level spacing statistics
• Visualisation/
  • BilliardVis.jl -- stationary solutions
  • SpacingVis.jl -- level spacing statistics
  • WeylComp.jl -- comparison to weyl prediction
  • BoundaryVis.jl -- boundary data

Installation

This package can be installed from Github using the the command:

using Pkg
Pkg.add(url="https://github.com/MatthewScargill/QuantumBilliards.jl")

Quickstart example

using QuantumBilliards

# Set the number of boundary discretization points
N = 700  

# Extract nodes, normals, weights, and billiard geometry data using a _info function
xs, ns, w, geom_data = QuantumBilliards.cardioid_info(N)

# Find resonant modes between 1 and 15, evaluated at 2000 interim points
spectrum = QuantumBilliards.resonant_modes(1.0, 15.0, 2000, xs, ns, w)

# Isolate a resonant mode and produce the associated bound state solution on the billiard
test_res = spectrum[7] # picking the seventh mode
QuantumBilliards.plot_billiard(xs, ns, w, test_res)

(a markdown document showing a complete example workflow as well as an optimisation history for the project can be found in rsc/docs/)

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Questions, extensions, and research discussions are welcome.