Bound states for systems with quartic energy-momentum dispersion

TL;DR

The study investigates bound states for 1D quasiparticles with quartic dispersion $E(p)\sim p^4$ in polynomial potentials, combining WKB methods and the Wentzel complex approach with higher-order and hyperasymptotic corrections. It derives a Bohr-Sommerfeld–type quantization for the double quartic problem $H=a^4 \hat p^4+b^4 x^4$ and validates the results against a universal Gaussian-basis variational calculation and an exactly solvable square-well model. A key finding is that bound-state wavefunctions exhibit nodes in the classically forbidden region, signaling a breakdown of the oscillation theorem for quartic dispersion, while the theorem remains valid in the classically allowed region. The work motivates extensions to higher-order dispersions and higher dimensions, with relevance to flat-band and Dirac-like materials where potential-energy effects dominate.

Abstract

Bound states and its energies for systems with the quartic energy-momentum dispersion $E(p) \sim p^4$ and polynomial potentials are studied using the Wentzel-Kramers-Brillouin (WKB) semiclassical approximation and the Wentzel complex method taking into account higher order WKB corrections. The obtained energies are compared with numerical values found by applying the variational approach utilizing the universal Gaussian basis. It is found that the wave functions of the ground and higher-energy states for systems with quartic dispersion have nodes in the classically forbidden region. Thus, the well-known oscillation theorem for the one-dimensional Schrödinger equation is not, in general, applicable for systems with quartic dispersion. Still it is observed that the oscillation theorem holds in the classically allowed region in all considered examples. The properties of bound state wave functions are compared with the solutions of the exactly solvable problem of a square well potential.

Figures (5)

• Figure 1: Contour enclosing two turning points.
• Figure 2: Left panel: The ground state wave function for the $H=\hat{p}^{4}+x^{4}$ (red solid line) and $H=\hat{p}^{4}+x^{2}$ Hamiltonians (blue dashed line). Middle panel: The wave functions of the first excited state for the $H=\hat{p}^{4}+x^{4}$ (red solid line) and $H=\hat{p}^{4}+x^{2}$ Hamiltonians (blue dashed line). Vertical red solid and blue dashed lines in the left and middle panels show boundaries of the classically allowed region for the ground and first excited states, respectively. Right panel: The wave function of the ground state (orange solid line) and the first excited state (brown dashed line) for the quartic oscillator with the Hamiltonian $H=\hat{p}^{2}+x^{4}$.
• Figure 3: The wave functions of the second (left panel), third (middle panel), and fourth excited state (right panel) for the potentials $V(x)=x^4$ (red solid lines) and $V(x)=x^2$ (blue dashed lines). Vertical red solid and blue dashed lines show boundaries of the corresponding classically allowed regions.
• Figure 4: Left panel: Energy levels $p L$ for positive and negative parity bound states as a function of $\lambda=v^{1/4}L$ (blue and red lines, respectively). Middle panel: energy levels as a function of the well width $L$ for the fixed well depth $v=2$. Right panel: energy levels as a function of the well depth $v$ for the fixed well width $L=1$.
• Figure 5: The wave function of the ground state (left panel) and the first excited state (right panel) for the potential well.