Abstract
This paper briefly summarizes previous work on complex classical mechanics and its relation to quantum mechanics. It then introduces a previously unstudied area of research involving the complex particle trajectories associated with elliptic potentials.
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The time evolution operator of such PT quantum systems has the usual form e −i Ht and this operator is unitary with respect to the Hermitian adjoint appropriate for the specific Hamiltonian H. Instead of the Dirac adjoint †, the adjoint for a PT-symmetric Hamiltonian H is given by CPT, where C is a linear operator satisfying the three simultaneous equations: C 2 = 1, [C,PT] = 0, and [C, H] = 0. The CPT norm is strictly real and positive and thus the theory is associated with a conventional Hilbert space. The time evolution is unitary because it preserves the CPT norms of vectors. A detailed discussion of these features of PT quantum mechanics is presented in [6].
The eigenfunctions ψ(x) of the Hamiltonian in (2.1) obey the differential equation −ψ″(x) + x 2(i x)ɛ ψ(x) = Eψ(x). These eigenfunctions are localized and decay exponentially in pairs of Stokes’ wedges in the complex-x plane, as is explained in S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998) [3]. The eigenfunctions live in L 2 but with the Dirac adjoint † replaced by the CPT adjoint. The eigenvalues are real and positive (see [7]). There is extremely strong numerical evidence that the set of eigenfunctions form a complete basis, but to our knowledge this result has not yet been rigorously established. Note that for the case ε = 2 the potential becomes −x 4, but in the complex plane this potential is not unbounded below! (The term unbounded below cannot be used in this context because the complex numbers are not ordered.) PT quantum mechanics has many qualitative features, such as arbitrarily fast time evolution, that distinguish it from conventional Dirac-Hermitian quantum mechanics. These features are discussed in [6].
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Bender, C.M., Hook, D.W., Kooner, K.S. (2011). Complex elliptic pendulum. In: Costin, O., Fauvet, F., Menous, F., Sauzin, D. (eds) Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation vol. I. Publications of the Scuola Normale Superiore, vol 12. Edizioni della Normale. https://doi.org/10.1007/978-88-7642-379-6_1
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