Spectral asymptotics of self-adjoint fourth order boundary value problems with eigenvalue parameter dependent boundary c
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RESEARCH
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Spectral asymptotics of self-adjoint fourth order boundary value problems with eigenvalue parameter dependent boundary conditions Manfred Möller* and Bertin Zinsou *
Correspondence: [email protected] The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
Abstract A regular fourth order differential equation with λ-dependent boundary conditions is considered. For four distinct cases with exactly one λ-independent boundary condition, the asymptotic eigenvalue distribution is presented. MSC: 34L20; 34B07; 34B08; 34B09 Keywords: fourth order boundary value problems; self-adjoint; boundary conditions; eigenvalue distribution; pure imaginary eigenvalues; spectral asymptotics
1 Introduction Sturm-Liouville problems have attracted extensive attention due to their intrinsic mathematical challenges and their applications in physics and engineering. However, apart from classical Sturm-Liouville problems, also higher order ordinary linear differential equations occur in applications, with or without the eigenvalue parameter in the boundary conditions. Such problems are realized as operator polynomials, also called operator pencils. Some recent developments of higher order differential operators whose boundary conditions depend on the eigenvalue parameter, including spectral asymptotics and basis properties, have been investigated in [–]. General characterizations of self-adjoint boundary conditions have been presented in [, ] for singular and (quasi-)regular problems. In all these cases, the minimal operator associated with an nth order differential equation must be symmetric, see [, ] for necessary and sufficient conditions. A more general discussion on the spectra of fourth order differential operators can be found in [, ]. The generalized Regge problem is realized by a second order differential operator which depends quadratically on the eigenvalue parameter and which has eigenvalue parameter dependent boundary conditions, see []. The particular feature of this problem is that the coefficient operators of this pencil are self-adjoint, and it is shown in [] that this gives some a priori knowledge about the location of the spectrum. In [] this approach has been extended to a fourth order differential equation describing small transversal vibrations of a homogeneous beam compressed or stretched by a force g. Separation of variables leads to a fourth order boundary problem with eigenvalue parameter dependent boundary © 2012 Möller and Zinsou; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Möller and Zinsou Boundary Value Problems 2012, 2012:106 http://www.boundaryvalueproblems.com/content/2012/1/106
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conditions, where the differential equa
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