Galactic Neutrino Communication

We examine the possibility to employ neutrinos to communicate within the galaxy. We discuss various issues associated with transmission and reception, and suggest that the resonant neutrino energy near 6.3 PeV may be most appropriate. In one scheme we propose to make Z^o particles in an overtaking e^+ - e^- collider such that the resulting decay neutrinos are near the W^- resonance on electrons in the laboratory. Information is encoded via time structure of the beam. In another scheme we propose to use a 30 PeV pion accelerator to create neutrino or anti-neutrino beams. The latter encodes information via the particle/anti-particle content of the beam, as well as timing. Moreover, the latter beam requires far less power, and can be accomplished with presently foreseeable technology. Such signals from an advanced civilization, should they exist, will be eminently detectable in neutrino detectors now under construction.Comment: 6 pages, 2 figures. Minor corrections, references adde

Reflectionless Potentials for Difference Schr\"odinger Equations

As a part of the program `discrete quantum mechanics,' we present general reflectionless potentials for difference Schr\"odinger equations with pure imaginary shifts. By combining contiguous integer wave number reflectionless potentials, we construct the discrete analogues of the $h(h+1)/\cosh^2x$ potential with the integer $h$, which belong to the recently constructed families of solvable dynamics having the $q$-ultraspherical polynomials with $|q|=1$ as the main part of the eigenfunctions. For the general ($h\in\mathbb{R}_{>0}$) scattering theory for these potentials, we need the connection formulas for the basic hypergeometric function ${}_2\phi_1(\genfrac{}{}{0pt}{}{a,b}{c}|q;z)$ with $|q|=1$, which is not known. The connection formulas are expected to contain the quantum dilogarithm functions as the $|q|=1$ counterparts of the $q$-gamma functions. We propose a conjecture of the connection formula of the ${}_2\phi_1$ function with $|q|=1$. Based on the conjecture, we derive the transmission and reflection amplitudes, which have all the desirable properties. They provide a strong support to the conjectured connection formula.Comment: 24 pages. Comments and references added. To appear in J. Phys.