This paper is an addendum to earlier papers cite{R1,R2} in which it was shown that the unstable separatrix solutions for Painleve I and II are determined by $PT$-symmetric Hamiltonians. In this paper unstable separatrix solutions of the fourth Painle
ve transcendent are studied numerically and analytically. For a fixed initial value, say $y(0)=1$, a discrete set of initial slopes $y(0)=b_n$ give rise to separatrix solutions. Similarly, for a fixed initial slope, say $y(0)=0$, a discrete set of initial values $y(0)=c_n$ give rise to separatrix solutions. For Painleve IV the large-$n$ asymptotic behavior of $b_n$ is $b_nsim B_{rm IV}n^{3/4}$ and that of $c_n$ is $c_nsim C_{rm IV} n^{1/2}$. The constants $B_{rm IV}$ and $C_{rm IV}$ are determined both numerically and analytically. The analytical values of these constants are found by reducing the nonlinear Painleve IV equation to the linear eigenvalue equation for the sextic $PT$-symmetric Hamiltonian $H=frac{1}{2} p^2+frac{1}{8} x^6$.
In a previous paper it was shown how to calculate the ground-state energy density $E$ and the $p$-point Greens functions $G_p(x_1,x_2,...,x_p)$ for the $PT$-symmetric quantum field theory defined by the Hamiltonian density $H=frac{1}{2}( ablaphi)^2+f
rac{1}{2}phi^2(iphi)^varepsilon$ in $D$-dimensional Euclidean spacetime, where $phi$ is a pseudoscalar field. In this earlier paper $E$ and $G_p(x_1,x_2,...,x_p)$ were expressed as perturbation series in powers of $varepsilon$ and were calculated to first order in $varepsilon$. (The parameter $varepsilon$ is a measure of the nonlinearity of the interaction rather than a coupling constant.) This paper extends these perturbative calculations to the Euclidean Lagrangian $L= frac{1}{2}( ablaphi)^2+frac{1}{2}mu^2phi^2+frac{1}{2} gmu_0^2phi^2big(imu_0^{1-D/2}phibig)^varepsilon-ivphi$, which now includes renormalization counterterms that are linear and quadratic in the field $phi$. The parameter $g$ is a dimensionless coupling strength and $mu_0$ is a scaling factor having dimensions of mass. Expressions are given for the one-, two, and three-point Greens functions, and the renormalized mass, to higher-order in powers of $varepsilon$ in $D$ dimensions ($0leq Dleq2$). Renormalization is performed perturbatively to second order in $varepsilon$ and the structure of the Greens functions is analyzed in the limit $Dto 2$. A sum of the most divergent terms is performed to {it all} orders in $varepsilon$. Like the Cheng-Wu summation of leading logarithms in electrodynamics, it is found here that leading logarithmic divergences combine to become mildly algebraic in form. Future work that must be done to complete the perturbative renormalization procedure is discussed.
This paper reports the results of an ongoing in-depth analysis of the classical trajectories of the class of non-Hermitian $PT$-symmetric Hamiltonians $H=p^2+ x^2(ix)^varepsilon$ ($varepsilongeq0$). A variety of phenomena, heretofore overlooked, have
been discovered such as the existence of infinitely many separatrix trajectories, sequences of critical initial values associated with limiting classical orbits, regions of broken $PT$-symmetric classical trajectories, and a remarkable topological transition at $varepsilon=2$. This investigation is a work in progress and it is not complete; many features of complex trajectories are still under study.
One-dimensional PT-symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form $H=p^2+V(x)$, where $V(x)$ is odd in $x$, pure imaginary, and vanishes as $|x|toinfty$. Five PT-symmetric po
tentials are studied: the Scarf-II potential $V_1(x)=iA_1,{rm sech}(x)tanh(x)$, which decays exponentially for large $|x|$; the rational potentials $V_2(x)=iA_2,x/(1+x^4)$ and $V_3(x)=iA_3,x/(1+|x|^3)$, which decay algebraically for large $|x|$; the step-function potential $V_4(x)=iA_4,{rm sgn}(x)theta(2.5-|x|)$, which has compact support; the regulated Coulomb potential $V_5(x)=iA_5,x/(1+x^2)$, which decays slowly as $|x|toinfty$ and may be viewed as a long-range potential. The real parameters $A_n$ measure the strengths of these potentials. Numerical techniques for solving the time-independent Schrodinger eigenvalue problems associated with these potentials reveal that the spectra of the corresponding Hamiltonians exhibit universal properties. In general, the eigenvalues are partly real and partly complex. The real eigenvalues form the continuous part of the spectrum and the complex eigenvalues form the discrete part of the spectrum. The real eigenvalues range continuously in value from $0$ to $+infty$. The complex eigenvalues occur in discrete complex-conjugate pairs and for $V_n(x)$ ($1leq nleq4$) the number of these pairs is finite and increases as the value of the strength parameter $A_n$ increases. However, for $V_5(x)$ there is an {it infinite} sequence of discrete eigenvalues with a limit point at the origin. This sequence is complex, but it is similar to the Balmer series for the hydrogen atom because it has inverse-square convergence.
PT-symmetric quantum mechanics began with a study of the Hamiltonian $H=p^2+x^2(ix)^varepsilon$. A surprising feature of this non-Hermitian Hamiltonian is that its eigenvalues are discrete, real, and positive when $varepsilongeq0$. This paper examine
s the corresponding quantum-field-theoretic Hamiltonian $H=frac{1}{2}( ablaphi)^2+frac{1}{2}phi^2(iphi)^varepsilon$ in $D$-dimensional spacetime, where $phi$ is a pseudoscalar field. It is shown how to calculate the Greens functions as series in powers of $varepsilon$ directly from the Euclidean partition function. Exact finite expressions for the vacuum energy density, all of the connected $n$-point Greens functions, and the renormalized mass to order $varepsilon$ are derived for $0leq D<2$. For $Dgeq2$ the one-point Greens function and the renormalized mass are divergent, but perturbative renormalization can be performed. The remarkable spectral properties of PT-symmetric quantum mechanics appear to persist in PT-symmetric quantum field theory.
This paper examines the complex trajectories of a classical particle in the potential V(x)=-cos(x). Almost all the trajectories describe a particle that hops from one well to another in an erratic fashion. However, it is shown analytically that there
are two special classes of trajectories x(t) determined only by the energy of the particle and not by the initial position of the particle. The first class consists of periodic trajectories; that is, trajectories that return to their initial position x(0) after some real time T. The second class consists of trajectories for which there exists a real time T such that $x(t+T)=x(t) pm2 pi$. These two classes of classical trajectories are analogous to valence and conduction bands in quantum mechanics, where the quantum particle either remains localized or else tunnels resonantly (conducts) through a crystal lattice. These two special types of trajectories are associated with sets of energies of measure 0. For other energies, it is shown that for long times the average velocity of the particle becomes a fractal-like function of energy.
This paper revisits earlier work on complex classical mechanics in which it was argued that when the energy of a classical particle in an analytic potential is real, the particle trajectories are closed and periodic, but that when the energy is compl
ex, the classical trajectories are open. Here it is shown that there is a discrete set of eigencurves in the complex-energy plane for which the particle trajectories are closed and periodic.
Suppose that a system is known to be in one of two quantum states, $|psi_1 > $ or $|psi_2 >$. If these states are not orthogonal, then in conventional quantum mechanics it is impossible with one measurement to determine with certainty which state the
system is in. However, because a non-Hermitian PT-symmetric Hamiltonian determines the inner product that is appropriate for the Hilbert space of physical states, it is always possible to choose this inner product so that the two states $|psi_1 > $ and $|psi_2 > $ are orthogonal. Thus, quantum state discrimination can, in principle, be achieved with a single measurement.
