On symplectic self-adjointness of Hamiltonian operator matrices

Symplectic self-adjointness of Hamiltonian operator matrices is studied, which is important to symplectic elasticity and optimal control. For the cases of diagonal domain and off-diagonal domain, necessary and sufficient conditions are shown. The proofs use Frobenius-Schur factorizations of unbounded operator matrices.Under additional assumptions, sufficient conditions based on perturbation method are obtained. The theory is applied to a problem in symplectic elasticity.     

Invariance of deficiency indices under perturbation for discrete Hamiltonian systems

This paper deals with discrete Hamiltonian systems with one singular endpoint. Using Hermitian linear relation generalized by linear Hamiltonian system, the invariance of the minimal and maximal deficiency indices under bounded perturbation for discrete Hamiltonian systems is built. This parallels the well-known results for linear Hamiltonian differential systems obtained by F.V. Atkinson.     

Boundedness of solutions for singular Hamiltonian differential systems with applications to deficiency indices

The paper gives boundedness estimation of solutions for singular Hamiltonian differential systems. As corollaries, limit-circle criteria are given and improve some previous results.     

Directed cyclic Hamiltonian cycle systems of the complete symmetric digraph

In this paper, we prove that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph, , exist if and only if n is odd with n≠15 and n≠p^α for p an odd prime and α≥2 or with n≠2p^α for p an odd prime and α≥1. We also show that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph minus a set of n/2 vertex-independent digons, (K_n−I)^∗, exist if and only if .     

The deficiency indices of a symmetric third-order differential operator

Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 468–471, September, 1995.     

Periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces

This paper discusses the existence and multiplicity of periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces. We prove that each such energy hypersurface carries at least one symmetric periodic orbit. Under some suitable pinching conditions, we also get an existence result of multiple symmetric periodic orbits.     

Defect indices and definiteness conditions for a class of discrete linear Hamiltonian systems

This paper is concerned with a class of discrete linear Hamiltonian systems in finite or infinite intervals. A definiteness condition and its equivalent statements are discussed and three sufficient conditions for the definiteness condition are given. A precise relationship between the defect index of the minimal subspace generated by the system and the number of linearly independent square summable solutions of the system is established. In particular, they are equal if and only if the definiteness condition is satisfied. Finally, two criteria for the limit point case and one criterion for the limit circle case are obtained.     

On diffusion in high-dimensional Hamiltonian systems

The purpose of this paper is to construct examples of diffusion for ε-Hamiltonian perturbations of completely integrable Hamiltonian systems in 2d-dimensional phase space, with d large.In the first part of the paper, simple and explicit examples are constructed illustrating absence of ‘long-time’ stability for size ε Hamiltonian perturbations of quasi-convex integrable systems already when the dimension 2d of phase space becomes as large as . We first produce the example in Gevrey class and then a real analytic one, with some additional work.In the second part, we consider again ε-Hamiltonian perturbations of completely integrable Hamiltonian system in 2d-dimensional space with ε-small but not too small, |ε|>exp(-d), with d the number of degrees of freedom assumed large. It is shown that for a class of analytic time-periodic perturbations, there exist linearly diffusing trajectories. The underlying idea for both examples is similar and consists in coupling a fixed degree of freedom with a large number of them. The procedure and analytical details are however significantly different. As mentioned, the construction in Part I is totally elementary while Part II is more involved, relying in particular on the theory of normally hyperbolic invariant manifolds, methods of generating functions, Aubry-Mather theory, and Mather's variational methods.Part I is due to Bourgain and Part II due to Kaloshin.     

On the self-adjointness of certain compact operators

It is shown that if the singular values of a compact operator T are dominated by those of the real part of T, then T must be self-adjoint. This improves an earlier result of Fong and Tsui. Some remarks on singular value inequalities associated with the Cartesian decomposition of a compact operator are also given.     

On the self-adjointness of certain compact operators

It is shown that if the singular values of a compact operator T are dominated by those of the real part of T, then T must be self-adjoint. This improves an earlier result of Fong and Tsui. Some remarks on singular value inequalities associated with the Cartesian decomposition of a compact operator are also given.     

On the essential self-adjointness of pseudodifferential operators

In this paper we investigate the self-adjointness for a kind of pseudodifferential operators, which include the nonsemi-bounded Schrödinger operator, ?Δ+v(x),v(x)→?∞, as │x│ → ∞, and the relativistic corrections to it, $\sqrt { - \Delta + m^2 } + v(x),v(x) \to - \infty ,as \left| x \right| \to \infty $ .     

On the interconnection structures of discretized port Hamiltonian systems

For numerical simulation and control design purposes, a mixed-finite element method [3] preserving the port Hamiltonian structure of the system has been developed [2]. This method was successfully applied for 1D systems. In this paper, we shall suggest some generalization of this result to higher dimensional spatial domain (3D) using Whitney forms as Galerkin base. The discretization procedure is illustrated on Maxwell's equations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the limit-point case of singular linear Hamiltonian systems

This article is concerned with the limit-point case (l.p.c.) of a Hamiltonian system. We present new proofs for several existing equivalent conditions on the l.p.c. established in terms of the asymptotic behaviour of the square integrable solutions of Hamiltonian systems with different spectral parameters and functions in the domain of the corresponding maximal operator, respectively. Further, we give two equivalent conditions in terms of the asymptotic behaviour of the square integrable solutions of Hamiltonian systems with the same complex and real spectral parameters, respectively. In addition, we establish two limit-point criteria which extend the relevant existing results.