Singular Dirac Systems and Sturm–Liouville Problems Nonlinear in the Spectral Parameter

Schmid, Harald; Tretter, Christiane

doi:10.1006/jdeq.2001.4082

Cited by 29 publications

(23 citation statements)

References 6 publications

(9 reference statements)

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“…These equations are considered e.g. in , , , . Remark The assumptions of Theorem 4.1 imply that the finite eigenvalues of (E) and (

\underset{̲}{E}

) are isolated and bounded from below, compare with [, Corollary 3.3 and Theorem 3.5]. Thus, if we denote by

- \infty < λ_{1} \leq λ_{2} \leq \dots \leq λ_{k} \leq \dots and - \infty < {\underset{̲}{λ}}_{1} \leq {\underset{̲}{λ}}_{2} \leq \dots \leq {\underset{̲}{λ}}_{k} \leq \dots

the finite eigenvalues of (E) and (

\underset{̲}{E}

), respectively, then inequality is equivalent with

{\underset{̲}{λ}}_{k} \leq λ_{k} for all k \in N,

whenever the k ‐th finite eigenvalues of (E) and (

\underset{̲}{E}

) exist.…”

Section: Comparison Of Finite Eigenvaluesmentioning

confidence: 99%

Comparison theorems for self‐adjoint linear Hamiltonian eigenvalue problems

Hilscher

2013

Mathematische Nachrichten

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In this work we derive new comparison results for (finite) eigenvalues of two self‐adjoint linear Hamiltonian eigenvalue problems. The coefficient matrices depend on the spectral parameter nonlinearly and the spectral parameter is present also in the boundary conditions. We do not impose any controllability or strict normality assumptions. Our method is based on a generalization of the Sturmian comparison theorem for such systems. The results are new even for the Dirichlet boundary conditions, for linear Hamiltonian systems depending linearly on the spectral parameter, and for Sturm–Liouville eigenvalue problems with nonlinear dependence on the spectral parameter.

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“…These equations are considered e.g. in , , , . Remark The assumptions of Theorem 4.1 imply that the finite eigenvalues of (E) and (

\underset{̲}{E}

) are isolated and bounded from below, compare with [, Corollary 3.3 and Theorem 3.5]. Thus, if we denote by

- \infty < λ_{1} \leq λ_{2} \leq \dots \leq λ_{k} \leq \dots and - \infty < {\underset{̲}{λ}}_{1} \leq {\underset{̲}{λ}}_{2} \leq \dots \leq {\underset{̲}{λ}}_{k} \leq \dots

the finite eigenvalues of (E) and (

\underset{̲}{E}

), respectively, then inequality is equivalent with

{\underset{̲}{λ}}_{k} \leq λ_{k} for all k \in N,

whenever the k ‐th finite eigenvalues of (E) and (

\underset{̲}{E}

) exist.…”

Section: Comparison Of Finite Eigenvaluesmentioning

confidence: 99%

Comparison theorems for self‐adjoint linear Hamiltonian eigenvalue problems

Hilscher

2013

Mathematische Nachrichten

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“…If ω = ω, p = p and ω or p change signs, one speaks about indefinite Sturm-Liouville problem. Quasi-derivatives related to the quasiderivatives generated by Shin-Zettl systems are used in the study of important modifications of Schrödinger-type operators (see, e.g., [13,49] and references therein) including Schrödinger-type operators with distributional potentials [13].…”

Section: Introductionmentioning

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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“…(ii) Note that the very same conditions on the coefficients r and q, namely positivity and monotonicity of r(t, ·) and strict monotonicity of q(t, ·) have been used in [11], [12], [16] in order to prove the accumulation of eigenvalues of (SL) at the boundary of the essential spectrum. The methods of this paper suggest that similar results also hold when q(t, ·) is only monotone (nondecreasing) in λ when the traditional notion of an eigenvalue is replaced by the notion of a finite eigenvalue (see Definition 3.1).…”

Section: Introductionmentioning

confidence: 99%

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

Böhner

Kratz

Hilscher

2012

Mathematische Nachrichten

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Key words Linear Hamiltonian system, self-adjoint eigenvalue problem, proper focal point, conjoined basis, finite eigenvalue, oscillation, controllability, normality, quadratic functional MSC (2010) 34L05, 34C10, 49N10, 93B60, 34L10In 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.The coefficients are n × n-matrix-valued functions such that the Hamiltoniandefined on [a, b] × R is symmetric, satisfies certain smoothness assumptions (see Section 2), and the monotonicity assumption

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Singular Dirac Systems and Sturm–Liouville Problems Nonlinear in the Spectral Parameter

Cited by 29 publications

References 6 publications

Comparison theorems for self‐adjoint linear Hamiltonian eigenvalue problems

Comparison theorems for self‐adjoint linear Hamiltonian eigenvalue problems

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

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

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