Singular Boundary Conditions for Sturm--Liouville Operators via Perturbation Theory

Bush, Michael D.; Frymark, Dale; Liaw, Constanze

doi:10.48550/arxiv.2011.03388

Cited by 2 publications

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“…The Friedrichs extension is usually defined through the closed semi-bounded form associated with a self-adjoint operator, see Section 4 (specifically equation ( 10)) for more about these forms or [6,9,30] for complete details. However, it suffices to think of the extension as the "smallest" self-adjoint extension among all other self-adjoint extensions (in the sense that it has the smallest form domain); it is often called the "soft" extension for this reason.…”

Section: A Complete Set Of Orthogonal Eigenfunctions Of the -Th Left-...mentioning

confidence: 99%

“…The difficulty of determining multiplicity of eigenvalues restricts the amount of evidence available to support the conjecture. In the case when A is the Jacobi differential operator, left-definite operators and Weyl -functions for their extensions can be found in [18] and the spectral analysis of [9] could be applied to potentially verify the conjecture for the example.…”

Section: Conjecturementioning

confidence: 99%

“…BKV theory was exploited in [9] to build a boundary pair for Sturm-Liouville operators with limit-circle endpoints that was compatible with a boundary triple. This allows the set up of a perturbation problem that describes all possible self-adjoint extensions of the minimal operator by using a scale of spaces in the following section.…”

Section: Proposition 1 Let a Be A Semi-bounded Self-adjoint Operator ...mentioning

confidence: 99%

“…Self-adjoint extensions of symmetric Sturm-Liouville operators with limit-circle endpoints were recently characterized in terms of a singular perturbation by the authors in [9]. The perturbation heavily exploited the connection between the scale of Hilbert spaces and BKV theory, along with boundary triples and boundary pairs (see e.g.…”

Section: Introductionmentioning

confidence: 99%

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Perspectives on General Left-Definite Theory

Frymark,

Liaw

2020

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In 2002, Littlejohn and Wellman developed a celebrated general leftdefinite theory for semi-bounded self-adjoint operators with many applications to differential operators. The theory starts with a semi-bounded self-adjoint operator and constructs a continuum of related Hilbert spaces and self-adjoint operators that are intimately related with powers of the initial operator. The development spurred a flurry of activity in the field that is still ongoing today.The main goal of this expository (with the exception of Proposition 1) manuscript is to compare and contrast the complementary theories of general left-definite theory, the Birman-Krein-Vishik (BKV) theory of self-adjoint extensions and singular perturbation theory. In this way, we hope to encourage interest in left-definite theory as well as point out directions of potential growth where the fields are interconnected. We include several related open questions to further these goals.

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Section: A Complete Set Of Orthogonal Eigenfunctions Of the -Th Left-...mentioning

confidence: 99%

Section: Conjecturementioning

confidence: 99%

Section: Proposition 1 Let a Be A Semi-bounded Self-adjoint Operator ...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Perspectives on General Left-Definite Theory

Frymark,

Liaw

2020

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“…For example, it is connected with Dirichlet-to-Neumann maps from partial differential equations. Apart from Bush-Frymark-Liaw [6], only scalar (i.e. not matrix-valued) spectral information was obtained and the connection to boundary conditions has not been worked out very explicitly.…”

Section: Introductionmentioning

confidence: 99%

Spectral Measures for Derivative Powers via Matrix-Valued Clark Theory

Bush,

Liaw,

Martin

2021

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The theory of finite-rank perturbations allows for the determination of spectral information for broad classes of operators using the tools of analytic function theory. In this work, finite-rank perturbations are applied to powers of the derivative operator, providing a full account from self-adjoint boundary conditions to computing aspects of the operators' matrix-valued spectral measures. In particular, the support and weights of the Clark (spectral) measures are computed via the connection between matrix-valued contractive analytic functions and matrix-valued nonnegative measures through the Herglotz Representation Theorem. For several powers of the derivative operator, explicit expressions are included. While eigenfunctions and eigenvalues for these operators with fixed boundary conditions can often be computed using direct methods from ordinary differential equations, this approach provides a more complete picture of the spectral information.

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Singular Boundary Conditions for Sturm--Liouville Operators via Perturbation Theory

Cited by 2 publications

References 0 publications

Perspectives on General Left-Definite Theory

Perspectives on General Left-Definite Theory

Spectral Measures for Derivative Powers via Matrix-Valued Clark Theory

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