Mathematical Physics

A problem of scattering by a Dirichlet right angle on a discrete square lattice is studied. The waves are governed by a discrete Helmholtz equation. The solution is looked for in the form of the Sommerfeld integral. The Sommerfeld transformant of the field is built as an algebraic function. The paper is a continuation of [1].

We consider quantum spins with S?? , and two-body interactions with O(2S+1) symmetry. We discuss the ground state phase diagram of the one-dimensional system. We give a rigorous proof of dimerization for an open region of the phase diagram, for S sufficiently large. We also prove the existence of a gap for excitations.

The direct and inverse scattering problems are analyzed for a first-order discrete system associated with the semi-discrete version of the derivative NLS system. The Jost solutions, the scattering coefficients, the bound-state dependency and norming constants are investigated and related to the corresponding quantities for two particular discrete linear systems associated with the semi-discrete version of the NLS system. The bound-state data set with any multiplicities is described in an elegant manner in terms of a pair of constant matrix triplets. Several methods are presented to the solve the inverse problem. One of these methods involves a discrete Marchenko system using as input the scattering data set consisting of the scattering coefficients and the bound-state information, and this method is presented in a way generalizable to other first-order systems both in the discrete and continuous cases for which a Marchenko system is not yet available. Finally, using the time-evolved scattering data set, the inverse scattering transform is applied on the corresponding semi-discrete derivative NLS system, and in the reflectionless case certain explicit solution formulas are presented in closed form expressed in terms of the two matrix triplets.

The two-dimensional Helmholtz equation separates in elliptic coordinates based on two distinct foci, a limit case of which includes polar coordinate systems when the two foci coalesce. This equation is invariant under the Euclidean group of translations and orthogonal transformations; we replace the latter by the discrete dihedral group of N discrete rotations and reflections. The separation of variables in polar and elliptic coordinates is then used to define discrete Bessel and Mathieu functions, as approximants to the well-known continuous Bessel and Mathieu functions, as N-point Fourier transforms approximate the Fourier transform over the circle, with integrals replaced by finite sums. We find that these 'discrete' functions approximate the numerical values of their continuous counterparts very closely and preserve some key special function relations.

This work builds on an existing model of discrete canonical evolution and applies it to the general case of a linear dynamical system, i.e., a finite-dimensional system with configuration space isomorphic to R q and linear equations of motion. The system is assumed to evolve in discrete time steps. The most distinctive feature of the model is that the equations of motion can be irregular. After an analysis of the arising constraints and the symplectic form, we introduce adjusted coordinates on the phase space which uncover its internal structure and result in a trivial form of the Hamiltonian evolution map. For illustration, the formalism is applied to the example of massless scalar field on a two-dimensional spacetime lattice.

For ???R 2 a connected, open, bounded set whose boundary is a finite union of polygons whose vertices have integer coordinates, the logarithm of the discrete Laplacian on L???Z 2 with Dirichlet boundary conditions has an asymptotic expansion for large L in which the term of order 1 is the logarithm of the zeta-regularized determinant of the corresponding continuum Laplacian. When ? is not simply connected, this result extends to Laplacians acting on two-valued functions with a specified monodromy class.

In this article we generalize the discrete Lagrangian and Hamiltonian mechanics on Lie groups to non-associative objects generalizing Lie groups (smooth loops). This shows that the associativity assumption is not crucial for mechanics and opens new perspectives. As a working example we obtain the discrete Lagrangian and Hamiltonian mechanics on unitary octonions.

The paper proves existence of renormalized stationary solutions for a dense class of discrete velocity Boltzmann equations in the plane with given ingoing boundary values. The proof is based on the construction of a sequence of approximations with L1 compactness for the integrated collision frequency and gain term. Compactness is obtained using the Kolmogorov-Riesz theorem.

We establish absolute continuity of the spectrum of a periodic Schrödiner operator in R^n with periodic perforations. We also prove analytic dependece of the dispersion relation on the shape of the perforation.

In this note we will discuss a potentially interesting extension of some recent results on primitive solutions to completely integrable partial differential equations. We will discuss a family distributions that are holomorphic on the Riemann sphere except on the singular sets homeomorphic to a Cantor set or Sierpinski gasket. These distributions allow us to produce solutions to the Kadomtsev--Petviashvili equation. These distributions are limits of families of rational functions that can also be associated with holomorphic line bundles on surfaces with a finite number of doubly degenerate singular points. We conjecture that a subset of these distributions can be used to formulate a definition of a holomorphic line bundle on some surfaces that are homeomorphic to spheres except where they become doubly degenerate on singular sets homeomorphic to a Cantor set or Sierpinski gasket.