Boundary conditions associated with the general left-definite theory for differential operators

Fleeman, Matthew; Frymark, Dale; Liaw, Constanze

doi:10.1016/j.jat.2018.10.005

Cited by 8 publications

(15 citation statements)

References 16 publications

(45 reference statements)

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“…Similar results for specific (mostly classical) operators can be found in [8,15,16] and their references. Some progress towards expressing left-definite spaces in terms of these standard boundary conditions was made much later in [29] and then expanded upon in [17].…”

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

confidence: 99%

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

confidence: 99%

Perspectives on General Left-Definite Theory

Frymark,

Liaw

2020

Preprint

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“…Here ∔ denotes the direct sum, and D + , D − are the defect spaces associated with the expression ℓ[ • ]. The decomposition can be made into an orthogonal direct sum by using the graph norm, see [21].…”

Section: Extension Theorymentioning

confidence: 99%

“…The associated sesquilinear form is defined, for f, g ∈ D J,n max , via equation (2.4). It can be written explicitly [21,Section 6] as…”

Section: Powers Of the Jacobi Differential Operatormentioning

confidence: 99%

“…The dependence of the functions ψ + j and ψ − j on the parameters α and β is suppressed here for simplicity. All of these functions are in the maximal domain [22,Lemma 4.3] and for j ≤ n they are also not in the minimal domain [21,Theorem 4.4]. The functions can also be used to define two finite-dimensional subspaces of D J,n max :…”

Section: Powers Of the Jacobi Differential Operatormentioning

confidence: 99%

“…The Corollary follows from Lemma 4.9 because the function u ± k is just a finite linear combination of v ± i functions, where i ≤ k. The functions u + k n k=1 and u − k n k=1 now produce a much simpler structure for the defect spaces when put into a matrix of sesquilinear forms, similar to that of[21, Equation 4.22], as compared to the starting families ψ + At the endpoint x = 1, observe …”

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confidence: 93%

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Boundary Triples and Weyl $m$-functions for Powers of the Jacobi Differential Operator

Frymark¹

2019

Preprint

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The abstract theory of boundary triples is applied to the classical Jacobi differential operator and its powers in order to obtain the Weyl m-function for several self-adjoint extensions with interesting boundary conditions: separated, periodic and those that yield the Friedrichs extension. These matrix-valued Nevanlinna-Herglotz m-functions are, to the best knowledge of the author, the first explicit examples to stem from singular higher-order differential equations.The creation of the boundary triples involves taking pieces, determined in [22], of the principal and non-principal solutions of the differential equation and putting them into the sesquilinear form to yield maps from the maximal domain to the boundary space. These maps act like quasi-derivatives, which are usually not well-defined for all functions in the maximal domain of singular expressions. However, well-defined regularizations of quasiderivatives are produced by putting the pieces of the non-principal solutions through a modified Gram-Schmidt process.

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Spectral Analysis, Model Theory and Applications of Finite-Rank Perturbations

Frymark

Liaw²

2020

Operator Theory, Operator Algebras and Their Interactions With Geometry and Topology

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Boundary conditions associated with the general left-definite theory for differential operators

Cited by 8 publications

References 16 publications

Perspectives on General Left-Definite Theory

Perspectives on General Left-Definite Theory

Boundary Triples and Weyl $m$-functions for Powers of the Jacobi Differential Operator

Spectral Analysis, Model Theory and Applications of Finite-Rank Perturbations

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