## Universal spectral behavior of $$x^2(ix)^ε$$ potentials

Carl M. Bender, Daniel W. Hook

The PT-symmetric Hamiltonian $$H=p^2+x^2(ix)^\epsilon$$ ($$\epsilon$$ real) exhibits a phase transition at $$\epsilon=0$$. When $$\epsilon\geq0$$ 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.

http://arxiv.org/abs/1205.4425
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)