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

Wi-Fi signals database construction using chebyshev wavelets for indoor positioning systems

Nowadays fast and accurate positioning of assets and people is as a crucial part of many businesses, such as, warehousing, manufacturing and logistics. Applications that offer different services based on mobile user location gaining more and more attention. Some of the most common applications include location-based advertising, directory assistance, point-to-point navigation, asset tracking, emergency and fleet management. While outdoors mostly covered by the Global Positioning System, there is no one versatile solution for indoor positioning. For the past decade Wi-Fi fingerprinting based indoor positioning systems gained a lot of attention by enterprises as an affordable and flexible solution to track their assets and resources more effectively. The concept behind Wi-Fi fingerprinting is to create signal strength database of the area prior to the actual positioning. This process is known as a calibration carried out manually and the indoor positioning system accuracy highly depends on a calibration intensity. Unfortunately, this procedure requires huge amount of time, manpower and effort, which makes extensive deployment of indoor positioning system a challenging task. approach of constructing signal strength database from a minimal number of measurements using Chebyshev wavelets approximation. The main objective of the research is to minimize the calibration workload while providing high positioning accuracy. The field tests as well as computer simulation results showed significant improvement in signal strength prediction accuracy compared to existing approximation algorithms. Furthermore, the proposed algorithm can recover missing signal values with much smaller number of on-site measurements compared to conventional calibration algorithm

Wi-Fi Signals Database Construction using Chebyshev Wavelets for Indoor Positioning Systems

Nowadays fast and accurate positioning of assets and people is as a crucial part of many businesses, such as, warehousing, manufacturing and logistics. Applications that offer different services based on mobile user location gaining more and more attention. Some of the most common applications include location-based advertising, directory assistance, point-to-point navigation, asset tracking, emergency and fleet management. While outdoors mostly covered by the Global Positioning System, there is no one versatile solution for indoor positioning. For the past decade Wi-Fi fingerprinting based indoor positioning systems gained a lot of attention by enterprises as an affordable and flexible solution to track their assets and resources more effectively. The concept behind Wi-Fi fingerprinting is to create signal strength database of the area prior to the actual positioning. This process is known as a calibration carried out manually and the indoor positioning system accuracy highly depends on a calibration intensity. Unfortunately, this procedure requires huge amount of time, manpower and effort, which makes extensive deployment of indoor positioning system a challenging task.  approach of constructing signal strength database from a minimal number of measurements using Chebyshev wavelets approximation. The main objective of the research is to minimize the calibration workload while providing high positioning accuracy.  The field tests as well as computer simulation results showed significant improvement in signal strength prediction accuracy compared to existing approximation algorithms. Furhtermore, the proposed algorithm can recover missing signal values with much smaller number of on-site measurements compared to conventional calibration algorithm

The Threshold effects for the two-particle Hamiltonians on lattices

For a wide class of two-body energy operators $h(k)$ on the three-dimensional lattice \bbZ^3, $k$ being the two-particle quasi-momentum, we prove that if the following two assumptions (i) and (ii) are satisfied, then for all nontrivial values $k$, $k\ne 0$, the discrete spectrum of $h(k)$ below its threshold is non-empty. The assumptions are: (i) the two-particle Hamiltonian $h(0)$ corresponding to the zero value of the quasi-momentum has either an eigenvalue or a virtual level at the bottom of its essential spectrum and (ii) the one-particle free Hamiltonians in the coordinate representation generate positivity preserving semi-groups

Wi-Fi received signal strength-based hyperbolic location estimation for indoor positioning systems

Nowadays, Wi-Fi fingerprinting-based positioning systems provide enterprises the ability to track their various resources more efficiently and effectively. The main idea behind fingerprinting is to build signal strength database of target area prior to location estimation. This process is called calibration and the positioning accuracy highly depends on calibration intensity. Unfortunately, calibration procedure requires huge amount of time and effort, and makes large scale deployments of Wi-Fi based indoor positioning systems non-trivial. In this research we present a novel location estimation algorithm for Wi-Fi based indoor positioning systems. The proposed algorithm combines signal sampling and hyperbolic location estimation techniques to estimate the location of mobile users. The algorithm achieves cost-efficiency by reducing the number of fingerprint measurements while providing reliable location accuracy. Moreover, it does not require any additional hardware upgrades to the existing network infrastructure. Experimental results show that the proposed algorithm with easy-to-build signal strength database performs more accurate than conventional signal strength-based methods

Solving system of nonlinear integral equations by Newton-Kantorovich method

Newton-Kantorovich method is applied to obtain an approximate solution for a system of nonlinear Volterra integral equations which describes a large class of problems in ecology, economics, medicine and other fields. The system of nonlinear integral equations is reduced to find the roots of nonlinear integral operator. This nonlinear integral operator is solved by the Newton-Kantorovich method with initial guess and this procedure is continued by iteration method to find the unknown functions. Finally, numerical examples are provided to show the validity and the efficiency of the method presented

On the Discrete Spectrum of a Model Operator in Fermionic Fock Space

We consider a model operator associated with a system describing three particles in interaction, without conservation of the number of particles. The operator acts in the direct sum of zero-, one-, and two-particle subspaces of the fermionic Fock space F ( 2 (T 3 )) over 2 (T 3 ). We admit a general form for the &quot;kinetic&quot; part of the Hamiltonian , which contains a parameter to distinguish the two identical particles from the third one. (i) We find a critical value * for the parameter that allows or forbids the Efimov effect (infinite number of bound states if the associated generalized Friedrichs model has a threshold resonance) and we prove that only for &lt; * the Efimov effect is absent, while this effect exists for any &gt; * . (ii) In the case &gt; * , we also establish the following asymptotics for the number ( ) of eigenvalues of below &lt; min = inf ess ( ) : lim → min ( ( )/| log | min − ||) = U 0 ( ) (U 0 ( ) &gt; 0), for all &gt; *