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Abstract: A technique for constructing an infinite tower of pairs of PT-symmetricHamiltonians, $\hat{H} n$ and $\hat{K} n$ n=2,3,4,

., that have exactly thesame eigenvalues is described. The eigenvalue problem for the first Hamiltonian$\hat{H} n$ of the pair must be posed in the complex domain, so itseigenfunctions satisfy a complex differential equation and fulfill homogeneousboundary conditions in Stokes- wedges in the complex plane. The eigenfunctionsof the second Hamiltonian $\hat{K} n$ of the pair obey a real differentialequation and satisfy boundary conditions on the real axis. This equivalenceconstitutes a proof that the eigenvalues of both Hamiltonians are real.Although the eigenvalue differential equation associated with $\hat{K} n$ isreal, the Hamiltonian $\hat{K} n$ exhibits quantum anomalies termsproportional to powers of $\hbar$. These anomalies are remnants of the complexnature of the equivalent Hamiltonian $\hat{H} n$. In the classical limit inwhich the anomaly terms in $\hat{K} n$ are discarded, the pair of Hamiltonians$H {n,classical}$ and $K {n,classical}$ have closed classical orbits whoseperiods are identical.

Author: Carl M. Bender, Daniel W. Hook

Source: https://arxiv.org/

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