On the method of finding periodic solutions of second-order neutral differential equations with piecewise constant arguments

This paper provides a method of finding periodical solutions of the second-order neutral delay differential equations with piecewise constant arguments of the form x″(t)+px″(t−1)=qx(2[t+12])+f(t), where [ ⋅ ] denotes the greatest integer function, p and q are nonzero constants, and f is a periodic function of t. This reduces the 2n-periodic solvable problem to a system of n+ 1 linear equations. Furthermore, by applying the well-known properties of a linear system in the algebra, all existence conditions are described for 2n-periodical solutions that render explicit formula for these solutions

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