Differential Operators and the Laguerre Type Polynomials

Everitt, W. N.; Krall, Allan M.; Littlejohn, Lance L.; Onyango-Otieno, Vitalis

doi:10.1137/0523037

Cited by 12 publications

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“…At the time of this writing, the other left-definite spaces H r and left-definite operators A r ðr > 0; r=1Þ associated with ðL 2 m ½À1; 1; AÞ are not explicitly known. We note that the first left-definite theory associated with the Laguerretype polynomials, which also satisfy a fourth-order Lagrangian symmetrizable differential equation, is also known; see [9], where the first left-definite space, its associated inner product, and the first left-definite operator are explicitly determined. Wellman [49] followed this work by analyzing, for each n 2 N 0 , the right-definite and first left-definite properties for the selfadjoint operator A ¼ AðnÞ, generated by the Laguerre-type differential equation of order 2n þ 4 (see [29]), having the generalized Laguerre-type polynomials as eigenfunctions.…”

Section: Further Examples and Concluding Remarksmentioning

confidence: 99%

“…Important results related to second-order equations have been obtained by Krall ([19,20,22,23]), Krall and Littlejohn [21], and Hajmirzaahmad ([16]). Left-definite results for higher-order differential equations have been obtained by Everitt and Littlejohn [11], Everitt et al [9,10,13,14], Loveland [30], and Wellman [49]. More recently, Vonhoff [48] has reconsidered the left-definite analysis of the fourth-order Legendre-type equation based on ideas developed in his thesis [47].…”

Section: Introductionmentioning

confidence: 95%

“…As we will see, each of the left-definite inner products can be seen as the Dirichlet inner product of the form (1.7) obtained from taking formal integral powers of the differential expression '½Á. Lastly, in Section 13, we outline a number of other applications (in particular, from [11] and [9]) and open problems resulting from this work.…”

Section: Introductionmentioning

confidence: 97%

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A General Left-Definite Theory for Certain Self-Adjoint Operators with Applications to Differential Equations

Littlejohn

Wellman

2002

Journal of Differential Equations

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DEDICATED TO W.N. EVERITT, AN INSPIRATION AND MENTOR TO BOTH AUTHORSWe show that any self-adjoint operator A (bounded or unbounded) in a Hilbert space H ¼ ðV ; ðÁ; ÁÞÞ that is bounded below generates a continuum of Hilbert spaces fH r g r>0 and a continuum of self-adjoint operators fA r g r>0 . For reasons originating in the theory of differential operators, we call each H r the rth left-definite space and each A r the rth left-definite operator associated with ðH ; AÞ. Each space H r can be seen as the closure of the domain DðA r Þ of the self-adjoint operator A r in the topology generated from the inner product ðA r x; yÞ ðx; y 2 DðA r ÞÞ. Furthermore, each A r is a unique self-adjoint restriction of A in H r . We show that the spectrum of each A r agrees with the spectrum of A and the domain of each A r is characterized in terms of another left-definite space. The Hilbert space spectral theorem plays a fundamental role in these constructions. We apply these results to two examples, including the classical Laguerre differential expression '½Á in which we explicitly find the leftdefinite spaces and left-definite operators associated with A, the self-adjoint operator generated by '½Á in L 2 ðð0; 1Þ; t a e Àt Þ having the Laguerre polynomials as eigenfunctions. # 2002 Elsevier Science (USA) Key Words: spectral theorem; self-adjoint operator; Hilbert space; Sobolev space; Dirichlet inner product; left-definite Hilbert space; left-definite self-adjoint operator; Laguerre polynomials; Stirling numbers of the second kind.Contents.

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Section: Further Examples and Concluding Remarksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 97%

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A General Left-Definite Theory for Certain Self-Adjoint Operators with Applications to Differential Equations

Littlejohn

Wellman

2002

Journal of Differential Equations

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“…Previous studies of fourth-order differential equations generating Legendre-type, Jacobi-type and Laguerretype orthogonal polynomials, see [3], [4] and [6], have shown that an initial study of the spectral properties of the differential equation in the classical Hilbert function space is essential to the subsequent study of spectral properties in the jump weighted Hilbert space.…”

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confidence: 99%

Additional spectral properties of the fourth‐order Bessel‐type differential equation

Everitt

Kalf

Littlejohn

et al. 2005

Mathematische Nachrichten

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Dedicated to the achievements and memory of F. V. Atkinson (1916Atkinson ( -2002 This paper discusses the spectral properties of the self-adjoint differential operator generated by the fourth-order Bessel-type differential expression, as defined by Everitt and Markett in 1994, in a Lebesgue-Stieltjes Hilbert function space. This space involves functions defined on the real line; the Lebesgue-Stieltjes measure is locally absolutely continuous on the real line, with the origin removed; the origin itself has strictly positive measure. It is shown that there is a unique such self-adjoint operator; this operator has no eigenvalues but has a continuous spectrum on the positive half-line of the spectral plane.

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Orthogonal Polynomials Satisfying Fourth Order Differential Equations

Krall¹

2002

Hilbert Space, Boundary Value Problems and Orthogonal Polynomials

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Differential Operators and the Laguerre Type Polynomials

Cited by 12 publications

References 11 publications

A General Left-Definite Theory for Certain Self-Adjoint Operators with Applications to Differential Equations

A General Left-Definite Theory for Certain Self-Adjoint Operators with Applications to Differential Equations

Additional spectral properties of the fourth‐order Bessel‐type differential equation

Orthogonal Polynomials Satisfying Fourth Order Differential Equations

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