This paper investigates Hamiltonian properties of the algebro-geometric discretization of KP hierarchy introduced in \cite{Gie1}. A Poisson bracket is introduced. The system is related to the periodic band matrix system of \cite{vM-M}. It is shown that the bracket descends to the latter and endows it with bi-Hamiltonian structure together with the first bracket already considered in \cite{vM-M}. On the other hand a bi-Hamiltonian structure for discrete KP seems to be absent for fundamental reasons. It is proven that the conserved quantities of both systems are in involution with respect to the bracket. A construction relating the bracket to a certain intersection pairing of cycles on a discrete torus is shown. This pairing is reminiscent of the intersection pairing in ``string topology'' \cite{C-S}.
We express discrete Painlevé equations as discrete Hamiltonian systems. The discrete Hamiltonian systems here mean the canonical transformations defined by generating functions. Our construction relies on the classification of the discrete Painlevé equations based on the surface-type. The discrete Hamiltonians we obtain are written in the logarithm and dilogarithm functions.
In this paper, we propose a discrete Hamilton--Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton--Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, we obtain the discrete equations for Hamiltonian dynamics by the discrete Legendre transformation. Based on the discrete contact Hamilton equation, we construct a discrete Hamilton--Jacobi equation for contact Hamiltonian dynamics. We show how the discrete Hamilton--Jacobi equation…
We consider the inverse scattering problems for two types of Schrödinger operators on locally perturbed periodic lattices. For the discrete Hamiltonian, the knowledge of the S-matrix for all energies determines the graph structure and the coefficients of the Hamiltonian. For locally perturbed equilateral metric graphs, the knowledge of the S-matrix for all energies determines the graph structure.
In this paper we introduce discrete gradient methods to discretize irreversible port-Hamiltonian systems showing that the main qualitative properties of the continuous system are preserved using this kind discretizations methods.
Under minimal regularity conditions on the photon dispersion and the coupling function, we prove that the spin-boson model with two massless photons in $\mathbb{R}^d$ cannot have more than two bound state energies whenever the coupling strength is sufficiently strong.
In this article, we relate the spectrum of the discrete magnetic Laplacian (DML) on a finite simple graph with two structural properties of the graph: the existence of a perfect matching and the existence of a Hamiltonian cycle of the underlying graph. In particular, we give a family of spectral obstructions parametrised by the magnetic potential for the graph to be matchable (i.e., having a perfect matching) or for the existence of a Hamiltonian cycle. We base our analysis on a special case of the spectral preorder introduced in [FCLP20a] and we use the magnetic potential as a spectral contro…
Variational integrators using forced discrete Hamiltonian systems
Universidad Nacional de Cuyo（门多萨国立大学）；Instituto Balseiro（巴尔塞罗研究所）；Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)（国家科学研究委员会）；Universidad Nacional de La Plata（拉普拉塔国立大学）；Facultad de Ingeniería Universidad Nacional de La Plata（拉普拉塔国立大学工程学院）；Centro de Matemática de La Plata (CMaLP)（拉普拉塔数学中心）；Facultad de Ciencias Exactas Universidad Nacional de La Plata（拉普拉塔国立大学理学院）
M. Kruskal showed that each continuous-time nearly-periodic dynamical system admits a formal $U(1)$ symmetry, generated by the so-called roto-rate. When the nearly-periodic system is also Hamiltonian, Noether's theorem implies the existence of a corresponding adiabatic invariant. We develop a discrete-time analoue of Kruskal's theory. Nearly-periodic maps are defined as parameter-dependent diffeomorphisms that limit to rotations along a $U(1)$-action. When the limiting rotation is non-resonant, these maps admit formal $U(1)$ symmetries to all orders in perturbation theory. For Hamiltonian near…
We use a Hamiltonian (transition matrix) description of height-restricted Dyck paths on the plane in which generating functions for the paths arise as matrix elements of the propagator to evaluate the length and area generating function for paths with arbitrary starting and ending points, expressing it as a rational combination of determinants. Exploiting a connection between random walks and quantum exclusion statistics that we previously established, we express this generating function in terms of grand partition functions for exclusion particles in a finite harmonic spectrum and present an …
We study spectral properties of two-dimensional canonical systems $y'(t)=zJH(t)y(t)$, $t\in[a,b)$, where the Hamiltonian $H$ is locally integrable on $[a,b)$, positive semidefinite, and Weyl's limit point case takes place at $b$. We answer the following questions explicitly in terms of $H$: Is the spectrum of the associated selfadjoint operator discrete ? If it is discrete, what is its asymptotic distribution ? Here asymptotic distribution means summability and limit superior conditions relative to comparison functions growing sufficiently fast. Making an analogy with complex analysis, this co…
Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and considering a stochastic generalization of the deterministic Lagrange-d'Alembert principle. Our approach presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators…
