Abstract
We consider eigenvalue problems for sixth-order ordinary differential equations. Such differential equations occur in mathematical models of vibrations of curved arches. With suitably chosen eigenvalue dependent boundary conditions, the problem is realized by a quadratic operator pencil. It is shown that the operators in this pencil are self-adjoint, and that the spectrum of the pencil consists of eigenvalues of finite multiplicity in the closed upper half-plane, except for finitely many eigenvalues on the negative imaginary axis.
Highlights
The spectral theory of higher order ordinary linear differential operators, in particular those with eigenvalue parameter dependent boundary conditions, is much less investigated and understood than the spectral theory of Sturm-Liouville operators
Like the spectral theory of Sturm-Liouville operators,regular and singular problems of higher order differential operators are distinguished by their spectral properties
Various aspects of higher order differential operators whose boundary conditions depend on the eigenvalue parameter, including spectral asymptotics and basis properties, have been investigated in [10, 11, 17, 28]
Summary
The spectral theory of higher order ordinary linear differential operators, in particular those with eigenvalue parameter dependent boundary conditions, is much less investigated and understood than the spectral theory of Sturm-Liouville operators. Various aspects of higher order differential operators whose boundary conditions depend on the eigenvalue parameter, including spectral asymptotics and basis properties, have been investigated in [10, 11, 17, 28]. Self-adjoint, boundary condition, eigenvalue, quadratic operator pencil. We will investigate the location of the spectrum of quadratic self-adjoint sixth-order differential operator pencils. Separation of variables leads to a sixth order ordinary differential equation y(6) + 2y(4) + y′′ = −cλ2(−y′′ + y), and the boundary conditions at a fixed end are such that y, y′ and y′′ are zero there, see [33, page 267] and [1, page 437], whereas the boundary conditions at the hinged end are such that y, y′ and y′ + y′′′ are zero, see [1, page 437]. The eigenvalues are located in the closed upper half-plane, with the possible exception of finitely many eigenvalues on the negative imaginary axis, inside an interval [0, −iν1/2], where ν is independent of α
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