690 research outputs found
Self-adjoint operators generated from non-Lagrangian symmetric differential equations having orthogonal polynomial eigenfunctions
No full textWe discuss the self-adjoint spectral theory associated with a certain fourth-order non-Lagrangian symmetrizable ordinary differential equation t(4)[y] = lambday that has a sequence of orthogonal polynomial solutions. This example was first discovered by Jung, Kwon, and Lee. In their paper, they derive the remarkable formula for these polynomials {Q(n)(x)}(n=0)infinity : Q(n)(x) = n integral(1)(x) PLn-1(t)dt, n is an element of N, where {PLn(x)}(n=0)(infinity) are the left Legendre type polynomials. The left Legendre type polynomials and the spectral analysis of the associated symmetric fourth-order differential equation that they satisfy have been extensively studied previously by Krall, Loveland, Everitt, and Littlejohn
Jacobi–Stirling numbers, Jacobi polynomials, and the left-definite analysis of the classical Jacobi differential expression
No full textAbstractWe develop the left-definite analysis associated with the self-adjoint Jacobi operator Ak(α,β), generated from the classical second-order Jacobi differential expressionℓα,β,k[y](t)=1wα,β(t)((-(1-t)α+1(1+t)β+1y′(t))′+k(1-t)α(1+t)βy(t))(t∈(-1,1)),in the Hilbert space Lα,β2(-1,1)≔L2((-1,1);wα,β(t)), where wα,β(t)=(1-t)α(1+t)β, that has the Jacobi polynomials {Pm(α,β)}m=0∞ as eigenfunctions; here, α,β>-1 and k is a fixed, non-negative constant. More specifically, for each n∈N, we explicitly determine the unique left-definite Hilbert–Sobolev space Wn,k(α,β)(-1,1) and the corresponding unique left-definite self-adjoint operator Bn,k(α,β) in Wn,k(α,β)(-1,1) associated with the pair (Lα,β2(-1,1),Ak(α,β)). The Jacobi polynomials {Pm(α,β)}m=0∞ form a complete orthogonal set in each left-definite space Wn,k(α,β)(-1,1) and are the eigenfunctions of each Bn,k(α,β). Moreover, in this paper, we explicitly determine the domain of each Bn,k(α,β) as well as each integral power of Ak(α,β). The key to determining these spaces and operators is in finding the explicit Lagrangian symmetric form of the integral composite powers of ℓα,β,k[·]. In turn, the key to determining these powers is a double sequence of numbers which we introduce in this paper as the Jacobi–Stirling numbers. Some properties of these numbers, which in some ways behave like the classical Stirling numbers of the second kind, are established including a remarkable, and yet somewhat mysterious, identity involving these numbers and the eigenvalues of Ak(α,β)
Singular point-like perturbations of the Laguerre operator in a Pontryagin space
No full textThe spectral problem for the Laguerre equation on (0, infinity) with real parameter a in the case 0 </alpha/ <1 is closely related to the Nevanlinna functionQ(alpha)(z) -piGamma(-z)/(sinpialpha)Gamma(-z -alpha ).If /alpha/ > 1 and /alpha/ not equal 2, 3,..., this function belongs to the generalized Nevanlinna class N-m, m = [/alpha/+1/2]. A natural question appears: to what spectral problem does this function correspond? For alpha <-1, alpha not equal -2, -3...., an answer was given by Derkach [D]. He obtained an operator representation for the function m(alpha)(Z) = -Q(alpha)(-z)/Gamma(2)(1 + alpha) in terms of a self-adjoint operator in a Pontyragin space. and an interpretation, of m. (z) as the Titchmarsh-Weyl function of some boundary value problem related to the Laguerre equation. That ail indefinite metric was needed was made clear earlier by Morton and Krall [MK]. In this note for alpha > 1, alpha not equal 2, 3... we answer this and related questions by using Pontryagin space operator realizations of suitable singular point-like perturbations of the Laguerre operator. We describe the operator models for Q (z) and compare them with the models for -alpha. Also we discuss the spectral properties of the self-adjoint linear relations in the representation of the functions Q(alpha)(z) and -Q(alpha)(z)(-1). Finally, we describe the connection between the self-adjoint linear relations in the representations of Q(alpha)(z) and Q(-alpha)(z+alpha) and show that this connection can be viewed as an operator implementation of the Kummer transform for confluent hypergeometric functions.</p
Singular point-like perturbations of the Laguerre operator in a Pontryagin space
No full textThe spectral problem for the Laguerre equation on (0, infinity) with real parameter a in the case 0 </alpha/ <1 is closely related to the Nevanlinna functionQ(alpha)(z) -piGamma(-z)/(sinpialpha)Gamma(-z -alpha ).If /alpha/ > 1 and /alpha/ not equal 2, 3,..., this function belongs to the generalized Nevanlinna class N-m, m = [/alpha/+1/2]. A natural question appears: to what spectral problem does this function correspond? For alpha <-1, alpha not equal -2, -3...., an answer was given by Derkach [D]. He obtained an operator representation for the function m(alpha)(Z) = -Q(alpha)(-z)/Gamma(2)(1 + alpha) in terms of a self-adjoint operator in a Pontyragin space. and an interpretation, of m. (z) as the Titchmarsh-Weyl function of some boundary value problem related to the Laguerre equation. That ail indefinite metric was needed was made clear earlier by Morton and Krall [MK]. In this note for alpha > 1, alpha not equal 2, 3... we answer this and related questions by using Pontryagin space operator realizations of suitable singular point-like perturbations of the Laguerre operator. We describe the operator models for Q (z) and compare them with the models for -alpha. Also we discuss the spectral properties of the self-adjoint linear relations in the representation of the functions Q(alpha)(z) and -Q(alpha)(z)(-1). Finally, we describe the connection between the self-adjoint linear relations in the representations of Q(alpha)(z) and Q(-alpha)(z+alpha) and show that this connection can be viewed as an operator implementation of the Kummer transform for confluent hypergeometric functions.</p
Toward monitoring and estimating the size of the HFO-contaminated seabed around a shipwreck using MBES backscatter data
No full textFunding Information: The authors are grateful to three anonymous reviewers for their constructive criticism and valuable comments. The authors express their gratitude to the Maritime Institute in Gdansk for making the final version of the report (Maritime Institute in Gdansk, 2016) available. The second author appreciates the support of the Gdynia Maritime University in funding this research through internal grant WN/PZ/2021/02. Funding Information: The second author appreciates the support of the Gdynia Maritime University in funding this research through internal grant WN/PZ/2021/02 . Publisher Copyright: © 2021Despite a progressive reduction of oil spills caused by the activity of maritime transportation, the latent sources of pollution still exist. Although the harmful impact of heavy fuel oil (HFO) on the marine environment is widely known, many shipwrecks cause contamination of the surrounding areas. In this paper, an approach to monitor the area of the HFO spill around a shipwreck is made using a bottom backscattering strength (BBS) obtained by a multibeam echosounder (MBES). As a case study, the s/s Stuttgart wreck located in the Gulf of Gdansk (Poland) is verified. Two different measurement campaigns have been carried out in shallow waters using low (190 kHz) and high (420 kHz) MBES frequency. The results indicate that the polluted area around s/s Stuttgart was estimated at 49.1 ha, which is around 18.3% more in comparison to the geological surveys made four years earlier.Peer reviewe
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