Theory Of Linear Hamiltonian Systems Research Articles

Generalized focal points and local Sturmian theory for linear Hamiltonian systems

Generalized focal points and local Sturmian theory for linear Hamiltonian systems

On the Eigenvalues of the Fermionic Angular Eigenfunctions in the Kerr Metric.

In view of a result recently published in the context of the deformation theory of linear Hamiltonian systems, we reconsider the eigenvalue problem associated with the angular equation arising after the separation of the Dirac equation in the Kerr metric, and we show how a quasilinear first order PDE for the angular eigenvalues can be derived efficiently. We also prove that it is not possible to obtain an ordinary differential equation for the eigenvalues when the role of the independent variable is played by the particle energy or the black hole mass. Finally, we construct new perturbative expansions for the eigenvalues in the Kerr case and obtain an asymptotic formula for the eigenvalues in the case of a Kerr naked singularity.

Renormalized Oscillation Theory for Linear Hamiltonian Systems on [0, 1] Via the Maslov Index

Working with a general class of regular linear Hamiltonian systems, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of \({\mathbb {C}}^{2n}\). We verify that our applicability class includes Dirac and Sturm–Liouville systems, as well as a system arising from differential-algebraic equations for which the spectral parameter appears nonlinearly.

Lidskii angles and Sturmian theory for linear Hamiltonian systems on compact interval

In this paper we investigate the Sturmian theory for general (possibly uncontrollable) linear Hamiltonian systems by means of the Lidskii angles, which are associated with a symplectic fundamental matrix of the system. In particular, under the Legendre condition we derive formulas for the multiplicities of the left and right proper focal points of a conjoined basis of the system, as well as the Sturmian separation theorems for two conjoined bases of the system, in terms of the Lidskii angles. The results are new even in the completely controllable case. As the main tool we use the limit theorem for monotone matrix-valued functions by Kratz (1993). The methods allow to present a new proof of the known monotonicity property of the Lidskii angles. The results and methods can also be potentially applied in the singular Sturmian theory on unbounded intervals, in the oscillation theory of linear Hamiltonian systems without the Legendre condition, in the comparative index theory, or in linear algebra in the theory of matrices.

Singular Sturmian separation theorems on unbounded intervals for linear Hamiltonian systems

In this paper we develop new fundamental results in the Sturmian theory for nonoscillatory linear Hamiltonian systems on an unbounded interval. We introduce a new concept of a multiplicity of a focal point at infinity for conjoined bases and, based on this notion, we prove singular Sturmian separation theorems on an unbounded interval. The main results are formulated in terms of the (minimal) principal solutions at both endpoints of the considered interval, and include exact formulas as well as optimal estimates for the numbers of proper focal points of one or two conjoined bases. As a natural tool we use the comparative index, which was recently implemented into the theory of linear Hamiltonian systems by the authors and independently by J. Elyseeva. Throughout the paper we do not assume any controllability condition on the system. Our results turn out to be new even in the completely controllable case.

Comparative index and Sturmian theory for linear Hamiltonian systems

The comparative index was introduced by J. Elyseeva (2007) as an efficient tool in matrix analysis, which has fundamental applications in the discrete oscillation theory. In this paper we implement the comparative index into the theory of continuous time linear Hamiltonian systems, study its properties, and apply it to obtain new Sturmian separation theorems as well as new and optimal estimates for left and right proper focal points of conjoined bases of these systems on bounded intervals. We derive our results for general possibly abnormal (or uncontrollable) linear Hamiltonian systems. The results turn out to be new even in the case of completely controllable systems. We also provide several examples, which illustrate our new theory.

Oscillation and spectral theory for linear Hamiltonian systems with nonlinear dependence on the spectral parameter

AbstractIn this paper, we consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. Our results generalize the known theory of linear Hamiltonian systems in two respects. Namely, we allow nonlinear dependence of the coefficients on the spectral parameter and at the same time we do not impose any controllability and strict normality assumptions. We introduce the notion of a finite eigenvalue and prove the oscillation theorem relating the number of finite eigenvalues which are less than or equal to a given value of the spectral parameter with the number of proper focal points of the principal solution of the system in the considered interval. We also define the corresponding geometric multiplicity of finite eigenvalues in terms of finite eigenfunctions and prove that the algebraic and geometric multiplicities coincide. The results are also new for Sturm–Liouville differential equations, being special linear Hamiltonian systems.

Multiple solutions of generalized asymptotical linear Hamiltonian systems satisfying Sturm–Liouville boundary conditions

This paper consider the multiple solutions for even Hamiltonian systems satisfying Sturm–Liouville boundary conditions. The gradient of Hamiltonian function is generalized asymptotically linear. The solutions obtained are shown to coincide with the critical points of a dual functional. Thanks to the index theory for linear Hamiltonian systems by Dong (2010) [1], we find critical points of this dual functional by verifying the assumptions of a lemma about multiple critical points given by Chang (1993) [2].

Sturmian theory for linear Hamiltonian systems without controllability

In this paper we present Sturmian separation and comparison theorems for linear Hamiltonian systems when no controllability assumption is imposed. This generalizes the traditional results of W. T. Reid for controllable (or normal) linear Hamiltonian systems to the possibly abnormal case. Our new theory is based on several recent results on linear Hamiltonian systems without controllability by W. Kratz, M. Wahrheit, V. Zeidan, and the author regarding the piecewise constant kernel for conjoined bases, the oscillation and eigenvalue theorems, and the Rayleigh principle. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Maslov-type index and brake orbits in nonlinear Hamiltonian systems

In this paper, we study the Maslov-type index theory for linear Hamiltonian systems with brake orbits boundary value conditions and its applications to the existence of multiple brake orbits of nonlinear Hamiltonian systems.

GKN theory for linear Hamiltonian systems

In this paper, we give the definition of maximal and minimal operators for linear Hamiltonian systems and investigate the relationship between the conjugate scalar product in a weighted Hilbert space and the skew-symmetric boundary form of the associated singular Hamiltonian operator, namely, the one-to-one correspondence between the set of self-adjoint extensions of the minimal operator and the set of Lagrangian symplectic subspaces. These results extend and improve the classical Glazman–Krein–Naimark (GKN) theory for quasi-differential operators.

P-index theory for linear Hamiltonian systems and multiple solutions for nonlinear Hamiltonian systems

For any P ∊ Sp(2n) and B ∊ L∞((0, 1);GLs(R2n)) we establish an index theory for a linear Hamiltonian system and discuss its applications in nonlinear Hamiltonian systems. First we will present two kinds of definitions by making use of the Maslov index and relative Morse index, respectively. Then we will prove the equivalence of these two definitions. Finally, we will make use of these index theories to discuss nontrivial solutions of nonlinear Hamiltonian systems.

Maslov type index theory for linear Hamiltonian systems with Bolza boundary value conditions and multiple solutions for nonlinear Hamiltonian systems

We study the classification of linear Hamiltonian systems satisfying Bolza boundary conditions and its applications to nonlinear Hamiltonian systems.

The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems

In this paper, the iteration formula of the Maslov-type index theory for linear Hamiltonian systems with continuous, periodic, and symmetric coefficients is established. This formula yields a new method to determine the minimality of the period for solutions of nonlinear autonomous Hamiltonian systems via their Maslov-type indices. Applications of this formula give new results on the existence of periodic solutions with prescribed minimal period for such systems, and unify known results under various convexity conditions.

Real transformations with polynomial invariants

This paper seeks to generalise one aspect of classical Krein theory for linear Hamiltonian systems by examining how the existence of a non-trivial, homogeneous, polynomial W of degree m ≥ 2 with <Ax, ▿W(x)&>; = 0,x ϵ R N , affects the spectrum of a real linear transformation A on R N . Amongst other things it is shown that (i) such a W exists if, and only if, the spectrum of A is linearly dependent over the natural numbers, and (ii) there exists such a W which is non-degenerate if, and only if, all the eigenvalues of A are imaginary and semi-simple. In classical Krein theory W is quadratic. Our enquiry is motivated by a theory of topological invariants for dynamical systems which have a first integral. Degenerate Hamiltonian systems are a special class where the present considerations are relevant.

Feedback Control for Linear Chaotic Systems

Abstract We study in a dynamical systems context the feedback stabilization problem for linear non-autonomous control processes with bounded measurable coefficients. we are led via standard methods to study the linear regulator problem, which we do by considering the spectral theory of linear Hamiltonian systems. This in turn is carried out via new methods involving the concepts of exponential dichotomy and rotation number. The feedback control matrix preserves recurrence properties and smoothness properties of the original coefficient matrices. If the coefficient matrices have a chaotic time dependence, then the feedback matrix will be “no m0re chaotic” than the coefficient matrices

Reversible linear systems and their versal deformations

Let R be a fixed linear involution (R 2=id) of the spaceR n. A linear operator M is said to bereversible with respect to R if RM R=M−1 and infinitesimally reversible with respect to R if M R=−RM. A linear differential equation dx/dt=B(t)x is said to be reversible with respect to R if V(t)R ≡−RV(−t). We construct normal forms and versal deformations for reversible and infinitesimally reversible operators. The results are applied to describe the homotopy classes of strongly stable reversible linear differential equations with periodic coefficients. The analogous theory for linear Hamiltonian systems was developed by J. Williamson, M. G. Krein, I.M. Gel'fand, V. B. Lidskii, D. M. Galin, and H. Koçak.