The spectral-bound conjecture for large skew-symmetric perturbations of the harmonic oscillator

Fix $k>0$ and let $f:\mathbb{R}\to\mathbb{R}$ be as in the paper. Let $H_\epsilon$ be the corresponding non-self-adjoint perturbation, let $\Sigma(\epsilon)$ denote its spectral bound, and let $\mu_0(\epsilon)$ and $\nu_0(\epsilon)$ be the eigenvalues specified by the asymptotic formulas

$$\mu_n(\epsilon)=\frac{i}{\epsilon}+(2n+1)\omega_\epsilon+\mathcal{O}(1),\qquad n=0,1,2,\dots,$$

and

$$\nu_n(\epsilon)=D_\epsilon+(2n+1)\Omega_\epsilon+\mathcal{O}(\epsilon^\nu),\qquad n=0,1,2,\dots.$$

Spectral-bound conjecture. For sufficiently small $\epsilon>0$,

$$\Sigma(\epsilon)=\min\{\operatorname{Re}(\mu_0(\epsilon)),\operatorname{Re}(\nu_0(\epsilon))\}.$$

This conjecture predicts that the spectral bound is determined by the lowest eigenvalue from one of the two asymptotic spectral families associated with the critical points of the complexified potential. The preceding analysis gives the quadratic approximations and formal eigenvalue asymptotics, but controlling the resolvent of the fully deformed non-self-adjoint operator is left for future work.