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

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Nihal Yokuş
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Differential Expression ◽  
Spectral Singularities ◽  
Sturm Liouville ◽  
Eigenvalue Dependent Boundary Conditions ◽  
Complex Valued

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 .

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

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Elgiz Bairamov ◽  
Nihal Yokus
Keyword(s):  
Boundary Conditions ◽  
Liouville Equation ◽  
Spectral Singularities ◽  
Sturm Liouville ◽  
Eigenvalue Dependent Boundary Conditions ◽  
Complex Valued

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.

scholarly journals Non-Self-Adjoint Singular Sturm-Liouville Problems with Boundary Conditions Dependent on the Eigenparameter

2010 ◽  
Vol 2010 ◽  
pp. 1-10
Author(s):  
Elgiz Bairamov ◽  
M. Seyyit Seyyidoglu
Keyword(s):  
Boundary Conditions ◽  
Finite Number ◽  
Analytic Functions ◽  
Liouville Problem ◽  
Uniqueness Theorems ◽  
Spectral Singularities ◽  
Sturm Liouville ◽  
Complex Valued

Let denote the operator generated in by the Sturm-Liouville problem: , , , where is a complex valued function and , with In this paper, using the uniqueness theorems of analytic functions, we investigate the eigenvalues and the spectral singularities of . In particular, we obtain the conditions on under which the operator has a finite number of the eigenvalues and the spectral singularities.

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

2015 ◽  
Vol 26 (10) ◽  
pp. 1550080 ◽  
Author(s):  
Esra Kir Arpat ◽  
Gökhan Mutlu
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Spectral Parameter ◽  
Spectral Properties ◽  
Hermitian Matrix ◽  
Boundary Value ◽  
Spectral Singularities ◽  
Sturm Liouville ◽  
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.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

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

2000 ◽  
Vol 130 (2) ◽  
pp. 239-247 ◽  
Author(s):  
P. A. Binding ◽  
P. J. Browne
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Separated Boundary Conditions ◽  
Sturm Liouville ◽  
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.

scholarly journals Isospectral sets for boundary value problems on the unit interval

1988 ◽  
Vol 8 (8) ◽  
pp. 301-358 ◽  
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Liouville Equation ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Unit Interval ◽  
Trace Function ◽  
Sturm Liouville ◽  
Real Self

AbstractWe analyse isospectral sets of potentials associated to a given ‘generalized periodic’ boundary condition in SL(2, R) for the Sturm-Liouville equation on the unit interval. This is done by first studying the larger manifold M of all pairs of boundary conditions and potentials with a given spectrum and characterizing the critical points of the map from M to the trace a + d Isospectral sets appear as slices of M whose geometry is determined by the critical point structure of the trace function. This paper completes the classification of isospectral sets for all real self-adjoint boundary conditions.

Dissipative Sturm-Liouville operators

1981 ◽  
Vol 88 (3-4) ◽  
pp. 329-343 ◽  
Author(s):  
Ian Knowles
Keyword(s):  
Hilbert Space ◽  
Differential Expression ◽  
Minimal Operator ◽  
Sturm Liouville ◽  
Image Position ◽  
Complex Valued ◽  
Liouville Operators

SynopsisConsider the differential expressionwherepandw> 0 are real-valued andqis complex-valued onI. A number of criteria are established for certain extensions of the minimal operator generated by τ in the weighted Hilbert spaceto be maximal dissipative.

scholarly journals An inverse spectral problem for Sturm-Liouville-type integro-differential operators with robin boundary conditions

2019 ◽  
Vol 50 (3) ◽  
pp. 207-221 ◽  
Author(s):  
Sergey Buterin
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Boundary Conditions ◽  
Differential Operators ◽  
A Priori ◽  
Sufficient Conditions ◽  
Robin Boundary Conditions ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Robin Boundary

The perturbation of the Sturm--Liouville differential operator on a finite interval with Robin boundary conditions by a convolution operator is considered. The inverse problem of recovering the convolution term along with one boundary condition from the spectrum is studied, provided that the Sturm--Liouville potential as well as the other boundary condition are known a priori. The uniqueness of solution for this inverse problem is established along with necessary and sufficient conditions for its solvability. The proof is constructive and gives an algorithm for solving the inverse problem.

scholarly journals INVESTIGATION OF SPECTRUM CURVES FOR A STURM-LIOUVILLE PROBLEM WITH TWO-POINT NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
Vol 25 (1) ◽  
pp. 37-52
Author(s):  
Kristina Bingelė ◽  
Agnė Bankauskienė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Characteristic Function ◽  
Boundary Conditions ◽  
Critical Points ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Point Boundary ◽  
Sturm Liouville

The article investigates the Sturm–Liouville problem with one classical and another nonlocal two-point boundary condition. We analyze zeroes, poles and critical points of the characteristic function and how the properties of this function depend on parameters in nonlocal boundary condition. Properties of the Spectrum Curves are formulated and illustrated in figures for various values of parameter ξ.

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