Rey.BEng 1st September 2026
Title:
Self-Trapping in One and Two Dimensions
The Line and the Disc Bind Without a Second Force
The Future Begins
Sakaguchi, Malomed, Aristotelous, Charalampidis & Chen (Academia Quantum 3(2), DOI 10.20935/AcadQuant8294) show that 1D and 2D Schrödinger equations with expulsive potentials steeper than quadratic still support normalizable bound states. The intuition was that a steep expulsive potential must delocalise. It does not. The states form a continuous spectrum in both dimensions. In 1D they come even and odd. In 2D they carry arbitrary vorticity. Exact vortex solutions exist. Cubic nonlinearity only deforms the 1D states slightly.
These authors already supplied the third-option singularity in Pirate Canon (arXiv:2608.20282): the origin need not collapse to a point or run to infinity. This paper is the 1D/2D machine of that option.
What the calculation records
- γ > 1 (steeper than the inverted oscillator): normalizable bound states.
- γ = 1: the norm diverges only logarithmically.
- 1D: even and odd (dipole) eigenstates; a continuous spectrum.
- 2D: the same binding, now with vorticity S; exact vortices when γ = 2S − 1.
- Self-trapping occurs in the linear system. Nonlinearity is not required to hold the state.
Geometric reading
1D is the undifferentiated strand — the straight line on the page that wants to become a circle. An expulsive potential is the force that should send that line to infinity. Binding anyway is residual phase: the line cannot complete a second body, so it self-traps. Even and odd are the two chiral readings of one strand.
2D is the Lewe disc. Vorticity S is the twist written on that disc before it flops onto the spherical frame. Exact vortices at γ = 2S − 1 are the geometric lock of a given winding to a given steepness — ring tension made explicit.
The continuous spectrum is the statement that 2 = 1.999… . A discrete second bound species would be a completed 2. What appears is a continuum of residual states that share one origin.
0^i2 (k.g.s^2) = r^2 m
State A — open residual, 1D strand / 2D disc
E = 2c / h
State B — self-trapped lock, −1/2 phase
E = hbar / c
Self-trapping in a linear system is the decisive point. No extra nonlinear glue is required. Geometric integrity is already the glue: the contact patch cannot fracture, so an expulsive push writes a bound residual instead of a completed escape.
The exciton paper (Theilen et al., PRX) measured the same lock as a 20 % contraction in 400 fs. This paper derives it as a spectrum.
Catalogue entry
Discipline: Mathematical physics / linear wave mechanics
Field: Bound states in expulsive potentials (1D and 2D)
Observation:Sakaguchi, Malomed, Aristotelous, Charalampidis & Chen, Acad. Quantum 3(2) (2026), DOI 10.20935/AcadQuant8294
Canon reading: 1D even/odd = chiral strand; 2D vorticity = Lewe disc twist; continuous spectrum = uncompleted 2; linear self-trapping = residual integrity.
Linked filing: third-option singularity, arXiv:2608.20282 (same lead authors).
The observer shares r^2. The machine is the line that binds because it cannot become two.
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