Exact isospectral pairs of {\cal P}{\cal T} symmetric Hamiltonians

Abstract

Abstract.A technique for constructing an infinite tower of pairs of PT -symmetric Hamiltonians,Ĥ n andK n (n = 2, 3, 4, . . .), that have exactly the same eigenvalues is described and illustrated by means of three examples (n = 2, 3, 4). The eigenvalue problem for the first HamiltonianĤ n of the pair must be posed in the complex domain, so its eigenfunctions satisfy a complex differential equation and fulfill homogeneous boundary conditions in Stokes' wedges in the complex plane. The eigenfunctions of the second HamiltonianK n of the pair obey a real differential equation and satisfy boundary conditions on the real axis. This equivalence constitutes a proof that the eigenvalues of both Hamiltonians are real. Although the eigenvalue differential equation associated withK n is real, the HamiltonianK n exhibits quantum anomalies (terms proportional to powers of ). These anomalies are remnants of the complex nature of the equivalent HamiltonianĤ n . For the cases n = 2, 3, 4 in the classical limit in which the anomaly terms inK n are discarded, the pair of Hamiltonians H n, classical and K n, classical have closed classical orbits whose periods are identical.

Exactly solvablePT-symmetric Hamiltonian having no Hermitian counterpart

“…Examples in which one can find exact expressions for the equivalent Dirac-Hermitian HamiltonianH in (3) associated with a given P T -symmetric Hamiltonian may be found in Refs. [12][13][14][15][16] and in our work in Ref. [5].…”

Quasi-exact solvability, resonances and trivial monodromy in ordinary differential equations

“…Hence, the norm (2.20) of the exactly-solvable states √ n|n is divergent and the corresponding radial eigenvalue problem is always 'irregular'. Higher-order differential equations of the form (5.1) have recently been studied in the context of PT symmetric quantum mechanics and its generalizations for even n in [20,21]. Additional motivation for the further study of higher-order eigenproblems of the type considered in this paper comes from their relevance to particular integrable quantum field theories, via the ODE/IM correspondence [19,13,[22][23][24].…”

Conduction bands in classical periodic potentials

“…Early work on the particle trajectories in complex classical mechanics is reported in refs [3,4]. Subsequently, detailed studies of the complex extensions of conventional classical-mechanical systems were undertaken: The remarkable properties of complex classical trajectories were examined in refs [5][6][7][8][9]; the complex behaviour of the pendulum, the LotkaVolterra equations for population dynamics and the Euler equations for rigid body rotation were studied in refs [10,11]; the complex Korteweg-de Vries equation was examined in refs [12][13][14][15]; the complex Riemann equation was examined in ref. [16]; complex Calogero models were examined in ref.…”