Universal spectral behavior of $x^2(ix{)}^{\epsilon }$ potentials

Abstract

The PT-symmetric Hamiltonian $H=p^2+x^2(ix{)}^{\epsilon }$ ($\epsilon$ real) exhibits a phase transition at $\epsilon =0$. When $\epsilon \ge 0$, the eigenvalues are all real, positive, discrete, and grow as $\epsilon$ increases. However, when $\epsilon <0$ there are only a finite number of real eigenvalues. As $\epsilon$ approaches -1 from above, the number of real eigenvalues decreases to one, and this eigenvalue becomes infinite at $\epsilon =-1$. In this paper it is shown that these qualitative spectral behaviors are generic and that they are exhibited by the eigenvalues of the general class of Hamiltonians $H^{(2n)}=p^{2n}+x^2(ix{)}^{\epsilon }$ ($\epsilon$ real, n=1, 2, 3, ...). The complex classical behaviors of these Hamiltonians are also examined.

8 pages, 7 figures

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Cited by in corpus (2)

• Real-Time Transport in Open Quantum Systems From $\mathcal{P}\mathcal{T}$-Symmetric Quantum Mechanics
• Complex Lagrangian dynamics