Analytic approximation for eigenvalues of a class of $\mathcal{PT}$-symmetric Hamiltonians

An analytical approximation for the eigenvalues of $\mathcal{PT}$-symmetric Hamiltonian $\mathsf{H}=-d^2/d{x}^2-{\left(ix\right)}^{\epsilon +2}$, $\epsilon >-1$ is developed via simple basis sets of harmonic-oscillator wave functions with variable frequencies and equilibrium positions. We demonstrate that our approximation provides high accuracy for any given energy level for all values of $\epsilon >-1$.

Figure 1

The dependence of the eigenvalues $E_n$ on the parameter $\epsilon$. The dashed lines are the exact numerical solutions via the Arnoldi iteration. The solid lines are the analytic approximation.

Figure 2

The dependence of the eigenvalues $E_n$, their real and imaginary parts on the parameter $\epsilon$, in the region $-1<\epsilon \le 0$. The dashed lines are the exact numerical solutions via the Arnoldi iteration. The solid lines are the analytic approximation via Eq. (19). After the point at $\epsilon ={\epsilon }_{\text{branching}\text{point}}$, where two energy levels coalesce they appear as complex conjugate pairs, whose real parts together with the real energy levels are plotted on the left pane and imaginary parts on the right pane, respectively.

Figure 3

The dependence of the real and imaginary parts of eigenvalues on the parameter $\epsilon$, in the region $-1.05\le \epsilon <-0.95$. The dashed lines are the exact numerical solutions via the Arnoldi iteration. The solid lines are the analytic approximation via Eq. (19) in which the states $(0,1)$, $(2,3),...$ were mixed.