Contributions to the Study of Oscillation Properties of the Solutions of Linear Differential Equations of the Second Order

Richardson, R. G. D.

doi:10.2307/2370485

Cited by 68 publications

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“…We remark that Example 1.1 was discussed by Richardson [18] for µ = (50π) 2 , and by several subsequent authors for different values of µ. It seems that [18] contains the first hint of a variational principle for eigenvalues of truly indefinite problems. This connection is explored in detail in [5].…”

Section: Introductionmentioning

confidence: 64%

A Variational Principle in Krein Space II

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Hryniv

Najman³

2005

Integr. equ. oper. theory

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Section: Introductionmentioning

confidence: 64%

A Variational Principle in Krein Space II

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Hryniv

Najman³

2005

Integr. equ. oper. theory

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“…Darboux Transformation for Hamiltonian Systems 549 equations, non-real eigenvalues are of interest (see [6,7,11,26,27,41] and references therein). Thus, the insertion (and the removal as an operation which is inverse to the insertion) of non-real eigenvalues is of interest as well.…”

Section: Transformed Systems: Inserted Eigenvaluesmentioning

confidence: 99%

Hamiltonian Systems and Sturm–Liouville Equations: Darboux Transformation and Applications

Sakhnovich

2017

Integr. Equ. Oper. Theory

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Abstract. We introduce GBDT version of Darboux transformation forHamiltonian and Shin-Zettl systems as well as for Sturm-Liouville equations (including indefinite Sturm-Liouville equations). These are the first results on Darboux transformation for general-type Hamiltonian and for Shin-Zettl systems. The obtained results are applied to the corresponding transformations of the Weyl-Titchmarsh functions and to the construction of explicit solutions of dynamical systems, of two-way diffusion equations and of indefinite Sturm-Liouville equations. The energy of the explicit solutions of dynamical systems is expressed (in a quite simple form) in terms of the parameter matrices of GBDT. The insertion of non-real eigenvalues into the spectrum of indefinite SturmLiouville operators is studied.Mathematics Subject Classification. Primary 34B24, 34L15, 47E05; Secondary 34B20, 74H05.

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“…In 1918, Richardson [24] stated that even the SLP consisting of (1.1) and the Dirichlet BC can have non-real eigenvalues. However, there is an important class of SLP's associated with (1.1), the so-called ''left-definite'' problems, whose eigenvalues are all real.…”

Section: Introductionmentioning

confidence: 99%

Left-Definite Sturm–Liouville Problems

Kong

Zettl

2001

Journal of Differential Equations

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Left-definite regular self-adjoint Sturm-Liouville problems, with either separated or coupled boundary conditions, are studied. We give an elementary proof of the existence of eigenvalues for these problems. For any fixed equation, we establish a sequence of inequalities among the eigenvalues for different boundary conditions and estimate the range of each eigenvalue as a function on the space of boundary conditions. Some of our results here yield an algorithm for numerically computing the eigenvalues of a left-definite problem with an arbitrary coupled boundary condition. Our inequalities imply that the well-known asymptotic formula for the eigenvalues in the separated case also holds in the coupled case. Moreover, we study the continuous and differentiable dependence of the eigenvalues of the general left-definite problem on all the parameters in its differential equation and boundary condition.

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Contributions to the Study of Oscillation Properties of the Solutions of Linear Differential Equations of the Second Order

Cited by 68 publications

References 0 publications

A Variational Principle in Krein Space II

A Variational Principle in Krein Space II

Hamiltonian Systems and Sturm–Liouville Equations: Darboux Transformation and Applications

Left-Definite Sturm–Liouville Problems

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