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

Böhner, Martin; Kratz, Werner; Hilscher, Roman Šimon

doi:10.1002/mana.201100172

Cited by 20 publications

(30 citation statements)

References 15 publications

(23 reference statements)

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“…More precisely, we provide examples illustrating the Downloaded 11/17/14 to 130.113.76. 6. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php fact that the conclusions of Theorem 2.1 do not hold when the rank of R 2 (t) changes.…”

mentioning

confidence: 75%

See 1 more Smart Citation

A Generalized Index Theorem for Monotone Matrix-Valued Functions with Applications to Discrete Oscillation Theory

Kratz¹,

Hilscher²

2013

SIAM J. Matrix Anal. & Appl.

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An index theorem is a tool for computing the change of the index (i.e., the number of negative eigenvalues) of a symmetric monotone matrix-valued function when its variable passes through a singularity. In 1995, the first author proved an index theorem in which a certain critical matrix coefficient is constant. In this paper, we generalize the above index theorem to the case when this critical matrix may be varying, but its rank, as well as the rank of some additional matrix, are constant. This includes as a special case the situation when this matrix has a constant image. We also show that the index theorem does not hold when the main assumption on constant ranks is violated. Our investigation is motivated by the oscillation theory of discrete symplectic systems and Sturm-Liouville difference equations with nonlinear dependence on the spectral parameter, which was recently developed by the second author and for which we obtain new oscillation theorems. Motivation.In 1995, the first author proved a result called an "index theorem," which shows how to compute the change of the index of a symmetric monotone matrix-valued function when its variable passes through a singularity; see [19, Theorem 2] or [20, Theorem 3.4.1], and for comparison also [24, Proposition 2.5]. By the index of a matrix we mean the number of its negative eigenvalues. This result has been utilized in many applications, in particular in the oscillation theory of SturmLiouville differential equations, linear Hamiltonian systems, and discrete symplectic systems. The relevant references are [20, sections 4.2, 5.2, 7.2] and [5, 10, 22, 24]. More precisely, the index theorem in [19, Theorem 2 and Corollary 2] or [20, Theorem 3.4.1 and Corollary 3.4.2] is the crucial auxiliary result for the continuous and discrete oscillation theory as presented in the references [20] (see the proof of Theorem 7.1.2 therein) and [10] (see the proof of Theorem 1 in section 4.4 therein).One of the key assumptions of this index theorem is that one of the considered coefficients is constant, i.e., it does not depend on t. In our main result (Theorem 2.1 below) this would be the matrix R 2 ≡ R 2 (t). It is the aim of this paper to let R 2 (·) depend on t and to analyze how this dependence can be incorporated, so that the assertion of the index theorem still holds. The assumption on the constancy of R 2 (·) implies, of course, limitations in the applications, as, e.g., in the above-mentioned references [20, Assumption (7.1.3)] and [10], where particularly the matrices B k do not depend on the spectral parameter λ. A similar requirement regarding the independence on λ in the coefficients B k (λ) was recently observed in [24, section 6.4]. On the *

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confidence: 75%

“…It is well known that discrete symplectic systems constitute a natural analogue of continuous time (differential) linear Hamiltonian systems; see [6,8,20,23]. Symplectic systems with the nonlinear dependence on λ, as in (S λ ), were recently introduced by the second author in [24] in connection with the eigenvalue problem with Dirichlet boundary conditions…”

mentioning

confidence: 99%

A Generalized Index Theorem for Monotone Matrix-Valued Functions with Applications to Discrete Oscillation Theory

Kratz¹,

Hilscher²

2013

SIAM J. Matrix Anal. & Appl.

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“…However, in [5] a certain strict normality assumption is needed to prove that the eigenvalues are isolated and bounded from below, which is not required in the present paper. Our results can also be regarded as a discrete time analogue of the corresponding theory in [9] for the continuous time linear Hamiltonian systems…”

Section: Introductionmentioning

confidence: 86%

Oscillation theorems for discrete symplectic systems with nonlinear dependence in spectral parameter

Hilscher

2012

Linear Algebra and its Applications

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In this paper we open a new direction in the study of discrete symplectic systems and Sturm-Liouville difference equations by introducing nonlinear dependence in the spectral parameter. We develop the notions of (finite) eigenvalues and (finite) eigenfunctions and their multiplicities, and prove the corresponding oscillation theorem for Dirichlet boundary conditions. The present theory generalizes several known results for discrete symplectic systems which depend linearly on the spectral parameter. Our results are new even for special discrete symplectic systems, namely for Sturm-Liouville difference equations, symmetric three-term recurrence equations, and linear Hamiltonian difference systems.

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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: 95%

“…Let

n \in N

be fixed. Let be given

n \times n

piecewise continuously differentiable

((), C_{p}^{1})

functions

A (\cdot, \cdot)

B (\cdot, \cdot)

C (\cdot, \cdot)

[a, b] \times R

such that

B (t, λ)

and

C (t, λ)

are symmetric,

B (t, λ) \geq 0 for all t \in [a, b], λ \in R,

and the Hamiltonian matrix

H (\cdot, \cdot)

defined in satisfies the following, see [, Assumption (2.1)]. There exist a partition

a = τ_{0} < \dots < τ_{m} = b

[a, b]

and a partition

- \infty < \dots < λ_{k} < λ_{k + 1} < \dots < \infty

double-struckR

with no finite accumulation point such that

scriptH

is continuous on

[τ_{i}, τ_{i + 1}] \times R

for every

i = 0, \dots, m - 1

\overset{H}{true}

is continuous on

…”

Section: Oscillation and Spectral Properties Of Linear Hamiltonian Symentioning

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

Cited by 20 publications

References 15 publications

A Generalized Index Theorem for Monotone Matrix-Valued Functions with Applications to Discrete Oscillation Theory

A Generalized Index Theorem for Monotone Matrix-Valued Functions with Applications to Discrete Oscillation Theory

Oscillation theorems for discrete symplectic systems with nonlinear dependence in spectral parameter

Comparison theorems for self‐adjoint linear Hamiltonian eigenvalue problems

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