Eigenvalue asymptotics for differential operators on graphs

Currie, Sonja; Watson, Bruce A.

doi:10.1016/j.cam.2004.11.038

Cited by 17 publications

(19 citation statements)

References 4 publications

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“…/ and c. , a/ coincide. This implies relations (16). Moreover, it is known ( [20]) that c. , a/ is an even entire function of exponential type having the form…”

Section: Lemma 31mentioning

confidence: 90%

“…Note that

scriptL

is a self‐adjoint operator in H (). Because all the edges are of finite length and the real function q ∈ L 2 (0, a ), the spectrum of

scriptL

is discrete, that is, it consists of eigenvalues with infinity as the only accumulation point.…”

Section: Spectral Problem and Differential Operatormentioning

confidence: 99%

“…This completes the proof. It is worth noting that the conditions (11)-(13), (16), and (17) are not only necessary but also sufficient for a union of sequences of real numbers n…”

Section: Lemma 31mentioning

confidence: 99%

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Inverse Sturm-Liouville spectral problem on symmetric star-tree

Didenko

Rozhenko

2013

Math. Meth. Appl. Sci.

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A spectral problem for the Sturm–Liouville equation on the edges of an equilateral regular star‐tree with the Dirichlet boundary conditions at the pendant vertices and Kirchhoff and continuity conditions at the interior vertices is considered. The potential in the Sturm–Liouville equation is a real–valued square summable function, symmetrically distributed with respect to the middle point of any edge. If {λj}is a sequence of real numbers, necessary and sufficient conditions for {λj}to be the spectrum of the problem under consideration are established. Copyright © 2013 John Wiley & Sons, Ltd.

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“…/ and c. , a/ coincide. This implies relations (16). Moreover, it is known ( [20]) that c. , a/ is an even entire function of exponential type having the form…”

Section: Lemma 31mentioning

confidence: 90%

“…Note that

scriptL

is a self‐adjoint operator in H (). Because all the edges are of finite length and the real function q ∈ L 2 (0, a ), the spectrum of

scriptL

is discrete, that is, it consists of eigenvalues with infinity as the only accumulation point.…”

Section: Spectral Problem and Differential Operatormentioning

confidence: 99%

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Inverse Sturm-Liouville spectral problem on symmetric star-tree

Didenko

Rozhenko

2013

Math. Meth. Appl. Sci.

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“…In [7] it was shown that a formally self-adjoint boundary value problem on a graph, with K edges, denoted e i for i = 1, . .…”

Section: The M-matrix and The Matrix Prüfer Anglementioning

confidence: 99%

“…In this paper we make use of the fact that a self-adjoint boundary value problem on a graph can be reformulated as a self-adjoint boundary value problem for a system on [0, 1] with separated boundary conditions, see [7]. The Titchmarsh-Weyl M-matrix is defined in Section 2, where, in addition, it is shown that the M-matrix exists and is well-defined.…”

Section: Introductionmentioning

confidence: 97%

M-matrix asymptotics for Sturm–Liouville problems on graphs

Currie

Watson

2008

Journal of Computational and Applied Mathematics

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We consider a system formulation for Sturm-Liouville operators with formally self-adjoint boundary conditions on a graph. An M-matrix associated with the boundary value problem is defined and related to the matrix Prüfer angle associated with the system boundary value problem, and consequently with the boundary value problem on the graph. Asymptotics for the M-matrix are obtained as the eigenparameter tends to negative infinity. We show that the boundary conditions may be recovered, up to a unitary equivalence, from the M-matrix and that the M-matrix is a Herglotz function. This is the first in a series of papers devoted to the reconstruction of the Sturm-Liouville problem on a graph from its M-matrix.

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On the Well-posedness of Evolutionary Equations on Infinite Graphs

Waurick

Kaliske

2012

Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations

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Eigenvalue asymptotics for differential operators on graphs

Cited by 17 publications

References 4 publications

Inverse Sturm-Liouville spectral problem on symmetric star-tree

Inverse Sturm-Liouville spectral problem on symmetric star-tree

M-matrix asymptotics for Sturm–Liouville problems on graphs

On the Well-posedness of Evolutionary Equations on Infinite Graphs

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