QuantumBilliards.jl is an open-source Julia library offering detailed numerical solutions to the two-dimensional interior Helmholtz eigenproblem; a key area within quantum chaos research. This set of tools computes spectra and associated eigenstates using both conventional solvers alongside accelerated methods like Vergini, Saraceno scaling, EBIM, Beyn’s contour integral method, and CORK. The University of Maribor team’s new Julia library simplifies calculations within quantum chaos research by providing tools to solve complex problems involving how energy levels behave inside two-dimensional shapes, known as ‘quantum billiards’.

The open-source software combines established methods with faster algorithms for computing key properties like spectra and associated states. This unified framework supports various billiard designs, including those with corners or multiple connections, easing exploration of chaotic systems at a fundamental level.

The University of Maribor team has released QuantumBilliards.jl, an open-source Julia library designed to tackle complex calculations in quantum chaos research. The software numerically solves the interior Helmholtz eigenproblem; imagining sound waves bouncing around inside an oddly shaped room helps illustrate this mathematical solution which aids understanding similar behaviour in confined quantum particles. The tool supports diverse billiard designs, from simple shapes with smooth curves to more intricate ones featuring corners or multiple connections, enabling detailed exploration of chaotic systems.

High Resolution Simulations Enable Access To Previously Unreachable Billiard System States

This threshold unlocks access to highly excited eigenstates important for statistical analysis and semiclassical investigations, which were computationally impossible before. QuantumBilliards.jl unifies diverse numerical techniques, basis expansion methods alongside accelerated eigensolvers like Vergini, Saraceno scaling, expanded boundary integral method (EBIM), Beyn’s contour integration, and CORK within a single framework designed specifically for two-dimensional Helmholtz problems.

The software allows exploration of complex billiard geometries including those with corners or multiple connections while accurately calculating eigenvalues, reconstructing eigenfunctions from normal derivatives, and computing Poincaré/Husimi phase space representations. Stadium billiards calculations successfully demonstrate this advance; eigenstates with wavenumbers of 74.1838018, 74.22509066, and 74.26339429 were accurately computed, a level previously inaccessible for detailed analysis. Beyond simple shapes, the framework’s flexibility extends to handle intricate designs featuring sharp angles or multiple pathways, offering a unified approach to eigenvalue calculation and generating Poincaré/Husimi phase space representations efficiently managing dense discretizations at high wavenumbers.

Numerical stability limits accurate wavefunction calculation within intricate billiard designs

While QuantumBilliards.jl expands computational reach for simulating quantum chaos, achieving truly accurate results remains challenging when dealing with complex geometries; accurately resolving features near corners or within multiply connected domains requires careful numerical treatment to avoid instability and overflow errors during calculations of the wavefunctions’ behaviour. The team relies on stable short-step Taylor recurrences alongside logarithmic product quadrature, techniques designed to manage exponentially growing terms in certain scenarios. However, precise evaluation across all regimes is inherently difficult. This library represents strong progress by offering a flexible tool for exploring quantum chaos across varied geometries.

The University of Maribor team’s new software provides an unprecedented ability to investigate quantum chaos within two-dimensional shapes. It consolidates conventional computational methods with faster algorithms, streamlining calculations previously limited by geometric complexity or numerical inefficiency; this approach allows scientists to calculate how particles behave when confined inside boundaries, known as ‘quantum billiards’. A valuable open-source resource, it will certainly support further development in understanding chaotic behaviours within confined quantum spaces and offers potential insights into the behaviour of more complex systems.

QuantumBilliards.jl is a newly developed Julia library that enables detailed investigation of quantum chaos in two-dimensional billiard geometries. The software combines established calculation techniques with accelerated eigensolvers like the CORK method and Beyn’s contour-integral method, allowing for computation of eigenvalues, such as those at wavenumbers of 74.1838018, 74.22509066, and 74.26339429, previously difficult to obtain. This tool provides researchers with a flexible framework for constructing complex shapes and computing properties like Poincaré, Husimi phase-space representations; it supports further development in understanding chaotic behaviours within confined spaces.

👉 More information
🗞 QuantumBilliards.jl: An extensible, user friendly, high-performance Julia library for the large scale computation of spectra and eigenstates of quantum billiards and related problems
✍️ Matic Orel and Črt Lozej (University of Maribor)
🧠 ArXiv: https://arxiv.org/abs/2610.32605

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