Boundary triples and Weyl m-functions for powers of the Jacobi differential operator

Frymark, Dale

doi:10.1016/j.jde.2020.05.032

Cited by 8 publications

(14 citation statements)

References 34 publications

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“…. , 4, are the same as in [22]. The Weyl m-function associated with the extension A ∞ can be written as…”

Section: Boundary Triple and Boundary Pair Constructionmentioning

confidence: 99%

“…The newly expanded perturbation setup is also utilized to gain new insight about the classical Jacobi differential equation in Section 4, and builds upon some of the recent results in [22]. Notably, the Weyl- m function for important self-adjoint extensions was determined in [22] and utilized as a starting point for the spectral analysis here.…”

Section: Introductionmentioning

confidence: 99%

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Singular boundary conditions for Sturm–Liouville operators via perturbation theory

Bush¹,

Frymark

Liaw³

2022

Can. J. Math.-J. Can. Math.

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We show that all self-adjoint extensions of semi-bounded Sturm-Liouville operators with limit-circle endpoint(s) can be obtained via an additive singular form bounded self-adjoint perturbation of rank equal to the deficiency indices, say d ∈ {1, 2}. This characterization generalizes the well-known analog for semi-bounded Sturm-Liouville operators with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written aswhere A 0 is a distinguished self-adjoint extension and Θ a self-adjoint linear relation in C d . The perturbation is singular in the sense that it does not belong to the underlying Hilbert space but is form bounded with respect to A 0 , i.e. it belongs to H −1 (A 0 ), with possible 'infinite coupling.' A boundary triple and compatible boundary pair for the symmetric operator are constructed to ensure the perturbation is well-defined and self-adjoint extensions are in a one-to-one correspondence with self-adjoint relations Θ. The merging of boundary triples with perturbation theory provides a more holistic view of the operator's matrix-valued spectral measures: identifying not just the location of the spectrum, but also certain directional information. As an example, self-adjoint extensions of the classical Jacobi differential equation (which has two limit-circle endpoints) are obtained and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.

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“…. , 4, are the same as in [22]. The Weyl m-function associated with the extension A ∞ can be written as…”

Section: Boundary Triple and Boundary Pair Constructionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Singular boundary conditions for Sturm–Liouville operators via perturbation theory

Bush¹,

Frymark

Liaw³

2022

Can. J. Math.-J. Can. Math.

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“…In every computed case the conjecture seems to hold but in the stated generality the status is unclear. Verification in the case where is the Jacobi differential operator can be found in [18,Cor. 5.1].…”

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%

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Perspectives on General Left-Definite Theory

Frymark,

Liaw

2020

Preprint

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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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Perspectives on General Left-Definite Theory

Frymark¹,

Liaw

2021

Operator Theory: Advances and Applications

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Boundary triples and Weyl m-functions for powers of the Jacobi differential operator

Cited by 8 publications

References 34 publications

Singular boundary conditions for Sturm–Liouville operators via perturbation theory

Singular boundary conditions for Sturm–Liouville operators via perturbation theory

Perspectives on General Left-Definite Theory

Perspectives on General Left-Definite Theory

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