$\cal{PT}$-Symmetric GUP-Modified Single- and Double-Well Quartic Oscillators

Abstract

We study the GUP-deformed quartic oscillator $H_η=p^2+ηp^4+x^4$ and distinguish its real-axis realization from complex-contour realizations. A contour chosen to remain in the decay sectors compatible with the undeformed quartic oscillator gives low-lying eigenvalues that approach the ordinary quartic spectrum as $η\to0^+$. This limiting-compatible contour does not, by itself, produce a low-level $\PT$-symmetry-breaking transition. To investigate exceptional points while retaining the same quartic leading behavior, we then introduce the explicitly $\PT$-symmetric extension $H_{η,a,g}=p^2+ηp^4+x^4-a x^2+\ii g x^3$ and fix $a=4$, for which the real part of the potential has a double-well form. Harmonic-oscillator-basis diagonalization with basis dimensions up to $N=100$ yields a converged exceptional-point branch: the critical coupling increases from $g_{\mathrm{EP}}\simeq0.10222364$ at $η=0$ to $g_{\mathrm{EP}}\simeq0.15603309$ at $η=0.1$. Across the transition, the two lowest real eigenvalues coalesce and continue as a complex-conjugate pair. The exceptional-point results belong to this stated double-well extension, not to the original single-well Hamiltonian alone.