Some problems of the qualitative Sturm-Liouville theory on a spatial network

Pokornyi, Yu. V.; Pryadiev, V. L.

doi:10.1070/rm2004v059n03abeh000738

Cited by 29 publications

(22 citation statements)

References 28 publications

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“…In parallel to the variational aspects of boundary-value problems on graphs studied here and on trees in [21], the work of Pokornyi and Pryadiev, and Pokornyi, Pryadiev and Al-Obeid, in [17] and [18], should be noted for the extension of Sturmian oscillation theory to second order operators on graphs. The idea of approximating the behaviour of eigenfunctions and eigenvalues for a boundary-value problem on a graph by the behaviour of associated problems on the individual edges, used here, was studied in the definite case in [2], [11] and [22].…”

Section: Introductionmentioning

confidence: 93%

Indefinite boundary value problems on graphs

Currie¹,

Watson²

2011

Operators and Matrices

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Abstract. We consider the spectral structure of indefinite second order boundary-value problems on graphs. A variational formulation for such boundary-value problems on graphs is given and we obtain both full and half-range completeness results. This leads to a max-min principle and as a consequence we can formulate an analogue of Dirichlet-Neumann bracketing and this in turn gives rise to asymptotic approximations for the eigenvalues.Mathematics subject classification (2010): 34B09, 34B45, 34L10, 34L20.

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Section: Introductionmentioning

confidence: 93%

Indefinite boundary value problems on graphs

Currie¹,

Watson²

2011

Operators and Matrices

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“…(3) η(x) = 1 (4) η(y) = 0 for all y that cannot be connected to x without passing through I 0 . Applying the Fundamental Theorem of Calculus to the lower side of (29) and invoking the equivalence of (18) and (19), we see that…”

Section: Landscape Upper Bounds On Tunneling Regions Using Agmon's Mmentioning

confidence: 99%

Localization and landscape functions on quantum graphs

Harrell

Maltsev

2019

Trans. Amer. Math. Soc.

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We discuss explicit landscape functions for quantum graphs. By a "landscape function" Υ(x) we mean a function that controls the localization properties of normalized eigenfunctions ψ(x) through a pointwise inequality of the formThe ideal Υ is a function that a) responds to the potential energy V (x) and to the structure of the graph in some formulaic way; b) is small in examples where eigenfunctions are suppressed by the tunneling effect; and c) relatively large in regions where eigenfunctions may -or may not -be concentrated, as observed in specific examples. It turns out that the connectedness of a graph can present a barrier to the existence of universal landscape functions in the high-energy régime, as we show with simple examples. We therefore apply different methods in different régimes determined by the values of the potential energy V (x) and the eigenvalue parameter E.

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“…For example, such operators are used in elasticity theory [4], they are also applied for description of electromagnetic waves [5] and nuclear structure [6]. Matrix Sturm-Liouville operators also generalize Sturm-Liouville operators on metric graphs (see [7][8][9][10][11][12] are references therein). The latter operators are often called quantum graphs.…”

Section: Introductionmentioning

confidence: 99%

Spectral analysis of the matrix Sturm–Liouville operator

Bondarenko

2019

Bound Value Probl

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The self-adjoint matrix Sturm-Liouville operator on a finite interval with a boundary condition in the general form is studied. We obtain asymptotic formulas for the eigenvalues and the weight matrices of the considered operator. These spectral characteristics play an important role in the inverse spectral theory. Our technique is based on analysis of analytic functions and on the contour integration in the complex plane of the spectral parameter. In addition, we adapt the obtained asymptotic formulas to the Sturm-Liouville operators on a star-shaped graph with two different types of matching conditions.

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Some problems of the qualitative Sturm-Liouville theory on a spatial network

Cited by 29 publications

References 28 publications

Indefinite boundary value problems on graphs

Indefinite boundary value problems on graphs

Localization and landscape functions on quantum graphs

Spectral analysis of the matrix Sturm–Liouville operator

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