A new preprint proposes a mathematical way to construct generalized coherent states in quantum billiards by extending the system’s eigenfunctions beyond the billiard boundary. The proposed state is built by projecting a minimal-dispersion wave packet onto the subspace formed by those continued eigenstates.
The document is an arXiv v1 preprint dated 20 Aug 2026. Its evidence consists of formal equations, analytical model cases and an illustrative numerical construction, rather than an independently validated physical result.
A different way to handle the edge
The paper focuses on how generalized coherent states can be defined when the wavefunctions associated with a billiard are continued outside it. The goal is to give a coherent-state construction that keeps the boundary in the mathematical description instead of treating the expansion as an interior calculation alone.
The key step is to switch from the differential Schrödinger equation to an equivalent Balian–Bloch integral equation. In this framework, the integral equation is used to formulate how a billiard eigenfunction continues beyond the boundary.
The continued wavefunctions have nonzero values outside the billiard. The authors report that projecting onto the continued eigenstates avoids singular boundary behavior in the expansion, although the study gives no quantitative comparison of boundary errors.
For numerical work, the paper presents a Balian–Bloch resolvent connected to a surface delta-prime term. The resolvent is the operator-based route the authors use to formulate construction in billiards of arbitrary shape, while the numerical implementation represents the operators as matrices.
What the examples show
In the illustrative numerical billiard, a state away from the boundary resembles a free Gaussian wave packet. A state near the boundary develops oscillations instead, giving the two locations visibly different wave patterns within the proposed construction.
The paper also treats solvable cases, including a one-dimensional infinite potential well and Coxeter billiards. For these models, it reports analytic expressions written with Jacobi and Riemann theta functions.
In the infinite-well case, the convolution of a Gaussian with a Jacobi-theta expression is represented by another theta-function expression. The initially localized packet is reported to delocalize as it evolves, while its motion remains periodic.
The exterior part of the continued wavefunction also raises an interpretive question. The physical meaning of those nonzero values depends on how observables are integrated over the billiard; the study does not show that the exterior continuation changes observables when the integration is restricted to the billiard.
A framework still awaiting tests
The central result is a formal mathematical construction, and no independent empirical validation is reported. The numerical route is presented as a way to build states for arbitrary shapes, but its performance and error metrics are not reported.
The numerical example is illustrative rather than a quantitative benchmark. The analysis reports no error bars or convergence measures, so it does not establish how accurately the procedure works across different billiard shapes or near their boundaries.
The manuscript also says that the coherent-state wavefunction generally needs an additional normalization coefficient. The closed-form results are limited to the stated solvable cases, while other shapes require numerical treatment.
Long-term localization remains unresolved. The authors describe localization as short-lived, especially for chaotic billiards, and identify localization in the classical limit as an open problem.
The work reports support from HSE University Basic Research project HSE-BR-2025-6, “Quantum devices and technologies of the new generation”.
Paper data and sources
Original title: Coherent states in quantum billiards constructed in the basis of the continued eigenfunctions
Authors: I. D. Burkov, S. S. Seidov
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text