Spectral properties of Sturm–Liouville system with eigenvalue-dependent boundary conditions

In this paper, we consider the boundary value problem [Formula: see text][Formula: see text] where λ is the spectral parameter and [Formula: see text] is a Hermitian matrix such that [Formula: see text] and αi ∈ ℂ, i = 0, 1, 2, with α2 ≠ 0. In this paper, we investigate the eigenvalues and spectral singularities of L. In particular, we prove that L has a finite number of eigenvalues and spectral singularities with finite multiplicities, under the Naimark and Pavlov conditions.

Sixth order differential operators with eigenvalue dependent boundary conditions

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.

Principal Functions of Non-Selfadjoint Sturm-Liouville Problems with Eigenvalue-Dependent Boundary Conditions

We consider the operator generated in by the differential expression , and the boundary condition , where is a complex-valued function and , with . In this paper we obtain the properties of the principal functions corresponding to the spectral singularities of .

Darboux transformations and the factorization of generalized Sturm–Liouville problems

A version of the Darboux transformation is explored for Sturm-Liouville problems with eigenvalue-dependent boundary conditions, from differential-equation and operator-theoretic viewpoints. Some of the literature on Darboux's transformation is related in a historical introduction.

Spectral Singularities of Sturm-Liouville Problems with Eigenvalue-Dependent Boundary Conditions

LetLdenote the operator generated inL2(R+)by Sturm-Liouville equation−y′′+q(x)y=λ2y,x∈R+=[0,∞),y′(0)/y(0)=α0+α1λ+α2λ2, whereqis a complex-valued function andαi∈ℂ,i=0,1,2withα2≠0. In this article, we investigate the eigenvalues and the spectral singularities ofLand obtain analogs of Naimark and Pavlov conditions forL.

Sturm–Liouville problems with non-separated eigenvalue dependent boundary conditions

Sturm–Liouville differential equations are studied under non-separated boundary conditions whose coefficients are first degree polynomials in the eigenparameter. Situations are examined where there are at most finitely many non-real eigenvalues and also where there are only finitely many real ones.

Closed-Form Transient Response of One-Dimensional Continua

Exact and closed-form transient response of general one-dimensional distributed dynamic systems subject to arbitrary external, initial and boundary disturbances is determined. Non-self-adjoint operators characterizing damping, gyroscopic and circulatory effects, and eigenvalue-dependent boundary conditions are considered. Through introduction of augmented operators, a closed-form modal expansion of the displacement and internal forces of the distributed system is derived. The eigenfunction expansion is realized in a spatial state-space formulation, which systematically yields exact eigensolutions, eigenfunction normalization coefficients, and modal coordinates. The proposed method is illustrated on a cantilever beam with end mass, damper and spring.