An arXiv preprint reports collapse suppression and creation of a ground state in a three-dimensional model that includes an attractive inverse-square potential and a self-repulsive cubic term. It describes what the equations do, not what has been measured in a physical system.
The work is a perspective theoretical analysis of idealized Schrödinger and Gross–Pitaevskii equations. It has no human or animal participants; the analyzed material consists of analytical expressions, asymptotic approximations, stationary solutions and numerical time-evolution simulations.
The paper compares numerical stationary solutions with analytical asymptotic or interpolating approximations. The comparisons track how the equations behave at the singular center or at large distances, where collapse, localization and normalizability become the key questions.
The modeled results split by setting
In the two-dimensional inverse-square analysis, cubic self-repulsion is reported to be insufficient to suppress collapse. With quintic self-repulsion, the model is reported to support ground and vortex states with a convergent norm. In plain language, the state’s mathematical norm remains finite rather than diverging.
Perturbed-evolution simulations are reported to support stability of the ground-state families in the quintic model. The inverse-square ground-state family is likewise reported as stable under perturbed evolution, consistent with the anti-Vakhitov–Kolokolov criterion used in the analysis.
The vortex analysis adds a model-specific threshold: the inverse-square strength must exceed a threshold for those states to exist. The reported vortex solutions are therefore conditional on the potential’s strength.
Localization in an outward-pushing potential
The preprint also examines steep expulsive potentials, described in the analysis as outward-pushing terms. It reports a full spectrum of effectively localized, normalizable bound states. In ordinary language, that means finite-norm states are reported across the spectrum examined, even in a setting described as expulsive.
At the inverted-harmonic-oscillator boundary, however, the reported normalization integral is weakly divergent rather than convergent. That boundary case is mathematically distinct from the steep-expulsive result, which is reported to have localized, normalizable states across a full spectrum.
The simulations show more than one outcome
The modeled one-dimensional self-attractive case showed a split outcome. A smaller-norm even ground state—an even state is mirror-symmetric in the coordinate—remained stable in the tested simulation. A larger-norm state instead showed spontaneous spatial-symmetry breaking, meaning the calculation no longer retained that mirror-symmetric pattern.
In the reported examples, the stable case had a norm of 1.73 and an E value of -0.9. The larger-norm case had a norm of 3.07 and an E value of -1.1. Those figures describe selected solutions in the model; they are not probabilities, measurements or estimates from a physical sample.
The contrast makes the paper’s use of “stability” conditional on the modeled case. The simulations support stability for the reported ground-state families, while the tested self-attractive example shows symmetry breaking at a different modeled state. The reported results therefore describe distinct model cases rather than a single outcome.
A theoretical result, not a laboratory finding
Taken together, the preprint offers theoretical predictions about collapse, localization and bound-state structure in singular-potential equations. Its central results are statements about idealized wave-function models, not demonstrations that collapse has been suppressed in a laboratory system or that a physical system would retain the reported states.
The authors present the calculations as motivation for experimental realization and further theoretical development. Whether the modeled ground states, vortices and full-spectrum localized states can be produced and maintained in a physical system remains outside what these equations and simulations establish.
The document is identified as an arXiv preprint marked v1. Its acknowledgments report collaborations and discussions but no funding source, and the authors declare no conflict of interest in the context of the paper.
Paper data and sources
Original title: Quantum-mechanical wave functions in singular potentials: linear and nonlinear states
Authors: Hidetsugu Sakaguchi, Boris A. Malomed
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text