Schr\"{o}dinger operators on lattices. The Efimov effect and discrete spectrum asymptotics

The Hamiltonian of a system of three quantum mechanical particles moving on the three-dimensional lattice $\Z^3$ and interacting via zero-range attractive potentials is considered. For the two-particle energy operator $h(k),$ with k\in \T^3=(-\pi,\pi]^3 the two-particle quasi-momentum, the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of $h(k)$ for $k\neq0$ is proven, provided that $h(0)$ has a zero energy resonance. The location of the essential and discrete spectra of the three-particle discrete Schr\"{o}dinger operator H(K), K\in \T^3 being the three-particle quasi-momentum, is studied. The existence of infinitely many eigenvalues of H(0) is proven. It is found that for the number $N(0,z)$ of eigenvalues of H(0) lying below $z<0$ the following limit exists \lim_{z\to 0-} \frac {N(0,z)}{\mid \log\mid z\mid\mid}=\cU_0 with \cU_0>0. Moreover, for all sufficiently small nonzero values of the three-particle quasi-momentum $K$ the finiteness of the number $N(K,\tau_{ess}(K))$ of eigenvalues of $H(K)$ below the essential spectrum is established and the asymptotics for the number $N(K,0)$ of eigenvalues lying below zero is given.Comment: 28 page

The threshold effects for a family of Friedrichs models under rank one perturbations

A family of Friedrichs models under rank one perturbations $h_{\mu}(p),$ $p \in (-\pi,\pi]^3$, $\mu>0,$ associated to a system of two particles on the three dimensional lattice $\Z^3$ is considered. We prove the existence of a unique eigenvalue below the bottom of the essential spectrum of $h_\mu(p)$ for all nontrivial values of $p$ under the assumption that $h_\mu(0)$ has either a threshold energy resonance (virtual level) or a threshold eigenvalue. The threshold energy expansion for the Fredholm determinant associated to a family of Friedrichs models is also obtained.Comment: 15 page

Theory of photon coincidence statistics in photon-correlated beams

The statistics of photon coincidence counting in photon-correlated beams is thoroughly investigated considering the effect of the finite coincidence resolving time. The correlated beams are assumed to be generated using parametric downconversion, and the photon streams in the correlated beams are modeled by two partially correlated Poisson point processes. An exact expression for the mean rate of coincidence registration is developed using techniques from renewal theory. It is shown that the use of the traditional approximate rate, in certain situations, leads to the overestimation of the actual rate. The error between the exact and approximate coincidence rates increases as the coincidence-noise parameter, defined as the mean number of uncorrelated photons detected per coincidence resolving time, increases. The use of the exact statistics of the coincidence becomes crucial when the background noise is high or in cases when high precision measurement of coincidence is required. Such cases arise whenever the coincidence-noise parameter is even slightly in excess of zero. It is also shown that the probability distribution function of the time between consecutive coincidence registration can be well approximated by an exponential distribution function. The well-known and experimentally verified Poissonian model of the coincidence registration process is therefore theoretically justified. The theory is applied to an on-off keying communication system proposed by Mandel which has been shown to perform well in extremely noisy conditions. It is shown that the bit-error rate (BER) predicted by the approximate coincidence-rate theory can be significantly lower than the actual BER obtained using the exact theory