We investigate the threshold effects self-adjoint bounded discrete Schrödinger-type operators in the Hilbert space of square-summable complex-valued functions defined on the \(d\)-dimensional lattice and establish the existence of eigenvalues and their asymptotics at coupling constant threshold.

In this work, we take advantage of two geometric aspects of phase space representations for optical systems. One aspect involves the geometric representation of the Wigner and the Husimi distribution function in terms of their graphs, hence their visualization, in phase space. The other aspect deals with the Wigner function as a link between Fourier optics and geometric optics. We apply the Wigner function and Husimi function to analyze and come up with representations of Bessel beams, a special class of propagating waves with properties that have been found useful in many applications.

For the Kepler problem, the Fock transform is the starting point of the quantization in the momentum representation, and its associated Moser regularization map is an imbedding of energy surfaces that yields the dynamic completion. Here, we first introduce a version of the Fock transform, with the normalization chosen so that the analytic formula for the resulting Moser regularization map exhibits a uniform dependence on the energy. Next, we provide a characterization for the Moser regularization map in terms of contact form and symmetries.

The notions of coKähler manifolds and 3-cosymplectic manifolds are odd-dimensional analogues of the ones of Kähler manifolds and hyperKähler manifolds, respectively. In this paper, we obtain reduction theorems of coKähler manifolds and 3-cosymplectic manifolds. We show that Kähler and coKähler (hyperKähler and 3-cosymplectic) reductions admit a natural compatibility with respect to “cylinder constructions”. We further prove the compatibility of Kähler and coKähler reductions with respect to the “mapping torus construction”.

In 1760 Euler proved the integrability of the planar problem of two fixed centers of gravitation. Later Jacobi separated the variables and integrated the equations of motion in terms of elliptic coordinates. In 1860 Köningsberger found solutions expressed by the four Jacobi's theta functions. These solutions contain complete Abelian integrals and are not convenient for use. We present effective solutions in terms of the classical Jacobi's elliptic functions sn, cn and dn. Their arguments depend linearly on some pseudo-time and on certain elliptic modulus. It is proved that, if the energy of the system is negative, there exist exactly ten topologically different types of regular solutions. We illustrate graphically and comment each of these ten cases.