A new realization of the fourth-order derivative Pais-Uhlenbeck oscillator is constructed. This realization possesses no states of negative norm and has a real energy spectrum that is bounded below. The key to this construction is the recognition that in this realization the Hamiltonian is not Dirac Hermitian. However, the Hamiltonian is symmetric under combined space reflection $P$ and time reversal $T$. The Hilbert space that is appropriate for this $PT$-symmetric Hamiltonian is identified and it is found to have a positive-definite inner product. Furthermore, the time-evolution operator is unitary.

In a recent paper Bender and Mannheim showed that the unequal-frequency fourth-order derivative Pais-Uhlenbeck oscillator model has a realization in which the energy eigenvalues are real and bounded below, the Hilbert-space inner product is positive definite, and time evolution is unitary. Central to that analysis was the recognition that the Hamiltonian $H_{\mathrm{PU}}$ of the model is $\mathcal{P}\mathcal{T}$ symmetric. This Hamiltonian was mapped to a conventional Dirac-Hermitian Hamiltonian via a similarity transformation whose form was found exactly. The present paper explores the equal-frequency limit of the same model. It is shown that in this limit the similarity transform that was used for the unequal-frequency case becomes singular and that $H_{\mathrm{PU}}$ becomes a Jordan-block operator, which is nondiagonalizable and has fewer energy eigenstates than eigenvalues. Such a Hamiltonian has no Hermitian counterpart. Thus, the equal-frequency $\mathcal{P}\mathcal{T}$ theory emerges as a distinct realization of quantum mechanics. The quantum mechanics associated with this Jordan-block Hamiltonian can be treated exactly. It is shown that the Hilbert space is complete with a set of nonstationary solutions to the Schrödinger equation replacing the missing stationary ones. These nonstationary states are needed to establish that the Jordan-block Hamiltonian of the equal-frequency Pais-Uhlenbeck model generates unitary time evolution.

We apply the conformal gravity theory to a sample of 111 spiral galaxies whose rotation curve data points extend well beyond the optical disk. With no free parameters other than galactic mass-to-light ratios, the theory is able to account for the systematics that is observed in this entire set of rotation curves without the need for any dark matter at all. In previous applications of the theory, a central role was played by a universal linear potential term $V(r)={\gamma }_0{c}^{2}r/2$ that is generated through the effect of cosmology on individual galaxies, with the coefficient ${\gamma }_0=3.06\times {10}^{-30}{\mathrm{cm}}^{-1}$ being of cosmological magnitude. Because the current sample is so big and encompasses some specific galaxies whose data points go out to quite substantial distances from galactic centers, we are able to identify an additional globally induced universal term in the data, a quadratic $V(r)=-\kappa c^2{r}^2/2$ term that is induced by inhomogeneities in the cosmic background. With $\kappa$ being found to be of magnitude $\kappa =9.54\times {10}^{-54}{\mathrm{cm}}^{-2}$, through study of the motions of particles contained within galaxies we are thus able to both detect the presence of a global de Sitter-like component and provide a specific value for its strength. Our study suggests that invoking dark matter may be nothing more than an attempt to describe global physics effects such as these in purely local galactic terms.

We present a conformal gravity fit to the 20 largest of a sample of 110 spiral galaxies. We identify the presence of a universal quadratic potential V$\kappa${}(r)=-$\kappa${}c2r2/2 with $\kappa${}=9.54\ifmmode\times\else\texttimes\fi{}10-54cm-2 induced by cosmic inhomogeneities. When V$\kappa${}(r) is taken in conjunction with both a universal linear potential V$\gamma${}0(r)=$\gamma${}0c2r/2 with $\gamma${}0=3.06\ifmmode\times\else\texttimes\fi{}10-30cm-1 generated by the homogeneous cosmic background and the contribution generated by the local luminous matter in galaxies, the theory then accounts for the rotation curve systematics observed in the entire 110 galaxies, without the need for any dark matter whatsoever. Our study suggests that using dark matter may be nothing more than an attempt to describe global effects in purely local galactic terms. With V$\kappa${}(r) being negative, galaxies can only support bound orbits up to distances of order $\gamma${}0/$\kappa${}=100kpc, with global physics imposing a limit on the size of galaxies.