Appropriate Inner Product for PT-Symmetric Hamiltonians

A Hamiltonian $H$ that is not Hermitian can still have a real and complete energy eigenspectrum if it instead is $PT$ symmetric. For such Hamiltonians three possible inner products have been considered in the literature, the $V$ norm, the $PT$ norm, and the $C$ norm. Here $V$ is the operator that implements $VHV^{-1}=H^{\dagger}$, the $PT$ norm is the overlap of a state with its $PT$ conjugate, and $C$ is a discrete linear operator that always exists for any Hamiltonian that can be diagonalized. Here we show that it is the $V$ norm that is the most fundamental as it is always chosen by the theory itself. In addition we show that the $V$ norm is always equal to the $PT$ norm if one defines the $PT$ conjugate of a state to contain its intrinsic $PT$ phase. We discuss the conditions under which the $V$ norm coincides with the $C$ operator norm, and show that in general one should not use the linear $C$ operator but for the purposes that it is used one can instead use the antilinear $PT$ operator itself.Comment: 10 pages,revtex4. Final version, to appear in Phys. Rev.

Microlensing, Newton-Einstein Gravity and Conformal Gravity

We discuss some implications of the current round of galactic dark matter searches for galactic rotation curve systematics and dynamics, and show that these new data do not invalidate the conformal gravity program of Mannheim and Kazanas which has been advanced as a candidate alternative to both the standard second order Newton-Einstein theory and the need for dark matter.Comment: Plain TeX, 4 pages. To appear in the proceedings of the Fifth Annual Astrophysics Conference in Marylan