The continuous spectrum of bound states in expulsive potentials

Abstract

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.

Figures (8)

• Figure 1: The continuous curve: a numerically found spatially even solution of Eq. (\ref{['1Dphi']}) with $g=0$, $\gamma =1$ (the anti-HO expulsive potential) and $E=0$. The dashed curve: the asymptotic approximation (\ref{['asympt']}) for the same solution, with fitting constants $\chi _{0}=\pi /8$ and $\varphi _{0}=1$.
• Figure 2: The continuous curve: a numerically found spatially even solution of Eq. (\ref{['1Dphi']}) with $g=0$, $\gamma =2$ (the quartic expulsive potential) and $E=0$. The dashed curve: the asymptotic approximation (\ref{['asympt']}) for the same case, with fitting constants $\chi _{0}=\pi /6$ and $\varphi _{0}=0.44$.
• Figure 3: In panels (a) and (b), the continuous curves represent numerically found spatially odd (dipole-mode) solutions of Eq. (\ref{['1Dphi']}) with $g=0$ and $E=0$, for $\gamma =1$ and $2$, respectively (cf. their spatially even counterparts in Figs. \ref{['fig1']} and \ref{['fig2']}). The dashed curves represent the corresponding asymptotic approximation (\ref{['asympt']}) with fitting constants $\chi _{0}=3\pi /8$, $\varphi _{0}=1.45$ in (a), and $\chi _{0}=\pi /3$, $\varphi _{0}=1.72$ in (b).
• Figure 4: The spatially even eigenstate produced by the numerical solution of Eq. (\ref{['1Dphi']}) with $g=0$, $\gamma =2$, and $E=-200$.
• Figure 5: The numerically found dependence of the coordinate $x_{\max }$, at which the wave function attains the largest value, on the eigenvalue $E<0$, for $g=0$ and $\gamma =2$. The dependence is plotted on the log-log scale, with the straight dashed line corresponding to the approximate relation (\ref{['1/4']}).