The continuous spectrum of bound states in expulsive potentials

On the contrary to the common intuition that a steep expulsive potential makes quantum states widely delocalized, we demonstrate that one- and two-dimensional (1D and 2D) Schrödinger equations, which include expulsive potentials that are \emph{steeper than the quadratic} (anti-harmonic-oscillator) ones, give rise to \emph{normalizable} (effectively localized) eigenstates. These states constitute full continuous spectra in the 1D and 2D cases alike. In 1D, these are spatially even and odd eigenstates. The 2D states may carry any value of the vorticity (alias magnetic quantum number). Asymptotic approximations for wave functions of the 1D and 2D eigenstates, valid far from the center, are derived analytically, demonstrating excellent agreement with numerically found counterparts. Special exact solutions for vortex states are obtained in the 2D case. These findings suggest an extension of the concept of bound states in the continuum, in quantum mechanics and paraxial photonics. Gross-Pitaevskii equations are considered as the nonlinear extension of the 1D and 2D settings. In 1D, the cubic nonlinearity slightly deforms the eigenstates, maintaining their stability. On the other hand, the quintic self-focusing term, which occurs in the photonic version of the 1D model, initiates the dynamical collapse of states whose norm exceeds a critical value.

💡 Research Summary

The paper investigates a counter‑intuitive class of quantum states that appear in one‑ and two‑dimensional Schrödinger equations with repulsive (expulsive) potentials steeper than the quadratic anti‑harmonic‑oscillator (anti‑HO). While a conventional anti‑HO ((U\propto -x^{2})) produces weakly delocalized, logarithmically divergent wavefunctions, the authors show that any potential of the form (-\frac12 x^{2\gamma}) with (\gamma>1) (e.g., quartic, sextic, etc.) yields normalizable eigenfunctions for every real eigenvalue (E). Consequently, the full continuous spectrum (-\infty<E<+\infty) is populated by bound‑like states that are nevertheless square‑integrable.

In the 1D linear case, an asymptotic WKB‑type analysis leads to the expression
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