The effect of variable change on oscillation and disconjugacy criteria with applications to spectral theory and asymptotic theory

Linear and nonlinear Hamiltonian systems are studied on time scales . We unify symplectic flow properties of discrete and continuous Hamiltonian systems. A chain rule which unifies discrete and continuous settings is presented for our so-called alpha derivatives on generalized time scales. This chain rule allows transformation of linear Hamiltonian systems on time scales under simultaneous change of independent and dependent variables, thus extending the change of dependent variables recently obtained by Došlý and Hilscher. We also give the Legendre transformation for nonlinear Euler-Lagrange equations on time scales to Hamiltonian systems on time scales.

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“…The following result generalizes Theorem 6.2 of [2], Section 3.18 of [4, p. 144], and Theorem 1 of Došlý and Hilscher [8]. …”

Section: Transformation Theorymentioning

confidence: 60%

Hamiltonian Systems on Time Scales

Ahlbrandt

Böhner

Ridenhour

2000

Journal of Mathematical Analysis and Applications

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“…The K.L. transformation for even order systems has been studied in great detail by Ahlbrandt, Hinton, and Lewis [2]. The use of the K.L.…”

Section: Introductionmentioning

confidence: 99%

“…One of the goals of this paper is the determination of the deficiency index def L 0 = (k, l) of L 0 . Since we assume L to be regular at a, i.e., the coefficients are locally integrable on [a, ∞), the only possibilities are (2, 1), (2,2), or (3, 3) as discussed in Section 2. In the case of maximal deficiency index any selfadjoint extension H of L 0 has a Hilbert Schmidt resolvent.…”

Section: Introductionmentioning

confidence: 99%

Deficiency indices and spectral theory of third order differential operators on the half line

Behncke

Hinton

2005

Mathematische Nachrichten

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We investigate the spectral theory of a general third order formally symmetric differential expression of the form. A Kummer-Liouville transformation is introduced which produces a differential operator unitarily equivalent to L. By means of the Kummer-Liouville transformation and asymptotic integration, the asymptotic solutions of L[y] = zy are found. From the asymptotic integration, the deficiency indices are found for the minimal operator associated with L. For a class of operators with deficiency index (2, 2), it is further proved that almost all selfadjoint extensions of the minimal operator have a discrete spectrum which is necessarily unbounded below. There are however also operators with continuous spectrum.

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“…Ahlbrandt, Hinton, and Lewis [1] performed both a dependent and independent variable change on continuous matrix operators of the form…”

Section: Introductionmentioning

confidence: 99%

“…Bohner and Došlý [3] applied this to a change of dependent variable on even order self-adjoint difference equations. We will use the method of [1] to perform a change of both dependent and independent variables on discrete matrix operators. This will enable us to perform a simultaneous change of dependent and independent variables on even order self-adjoint difference operators and on linear Hamiltonian difference systems.…”

Section: Introductionmentioning

confidence: 99%

Discrete Variable Transformations on Symplectic Systems and Even Order Difference Operators

Voepel

1998

Journal of Mathematical Analysis and Applications

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Change of independent variable t = 1/x motivates variable step size discretizations of even order differential operators. We develop variable change methods for discrete symplectic (i.e., J-orthogonal) systems. This enables us to perform simultaneous change of independent and dependent variables on discrete linear Hamiltonian systems and on newly defined even order variable step size formally self adjoint difference operators. These variable changes yield a new system which is related to the original system by an operator identity. We generalize results of Bohner and Došlý on transformations of formally self-adjoint scalar difference operators. They only considered a change of dependent variable whereas these methods allow y x n = µ x n z t n where t n = f x n These variable change results bring the subject of transformation theory for even order difference operators closer to the known transformation theory in the continuous case.

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The effect of variable change on oscillation and disconjugacy criteria with applications to spectral theory and asymptotic theory

Cited by 49 publications

References 49 publications

Hamiltonian Systems on Time Scales

Hamiltonian Systems on Time Scales

Deficiency indices and spectral theory of third order differential operators on the half line

Discrete Variable Transformations on Symplectic Systems and Even Order Difference Operators

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