scholarly journals Finite spectrum of Sturm-Liouville problems with eigenparameter-dependent boundary conditions on time scales

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1747-1757
Author(s):  
Ji-Jun Ao ◽  
Juan Wang
Keyword(s):  
Spectral Analysis ◽  
Boundary Conditions ◽  
Time Scales ◽  
Liouville Equation ◽  
Time Scale ◽  
The Self ◽  
Finite Spectrum ◽  
Sturm Liouville

The spectral analysis of a class of Sturm-Liouville problems with eigenparameter-dependent boundary conditions on bounded time scales is investigated. By partitioning the bounded time scale such that the coefficients of Sturm-Liouville equation satisfy certain conditions on the adjacent subintervals, the finite eigenvalue results are obtained. The results show that the number of eigenvalues not only depend on the partition of the bounded time scale, but also depend on the eigenparameter-dependent boundary conditions. Both of the self-adjoint and non-self-adjoint cases are considered in this paper.

scholarly journals Spectral analysis of singular Sturm-Liouville operators on time scales

2018 ◽  
Vol 72 (1) ◽  
pp. 1 ◽  
Author(s):  
Bilender P. Allahverdiev ◽  
Huseyin Tuna
Keyword(s):  
Spectral Analysis ◽  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Time Scales ◽  
Liouville Operator ◽  
Adjoint Operator ◽  
The Self ◽  
Sturm Liouville ◽  
Liouville Operators

In this paper, we consider properties of the spectrum of a Sturm-Liouville<br />operator on time scales. We will prove that the regular symmetric<br />Sturm-Liouville operator is semi-bounded from below. We will also give some<br />conditions for the self-adjoint operator associated with the singular<br />Sturm-Liouville expression to have a discrete spectrum. Finally, we will<br />investigate the continuous spectrum of this operator.

scholarly journals Geometric aspects of self-adjoint Sturm–Liouville problems

2017 ◽  
Vol 147 (6) ◽  
pp. 1279-1295
Author(s):  
Yicao Wang
Keyword(s):  
Boundary Conditions ◽  
Liouville Equation ◽  
Adjoint Action ◽  
Principal Type ◽  
Unitary Matrices ◽  
Adjoint Orbits ◽  
Sturm Liouville ◽  
Refined Classification ◽  
Adjoint Orbit

In this paper we use U(2), the group of 2 × 2 unitary matrices, to parametrize the space of all self-adjoint boundary conditions for a fixed Sturm–Liouville equation on the interval [0, 1]. The adjoint action of U(2) on itself naturally leads to a refined classification of self-adjoint boundary conditions – each adjoint orbit is a subclass of these boundary conditions. We give explicit parametrizations of those adjoint orbits of principal type, i.e. orbits diffeomorphic to the 2-sphere S2, and investigate the behaviour of the nth eigenvalue λnas a function on such orbits.

scholarly journals STURM–LIOUVILLE PROBLEMS WITH BOUNDARY CONDITIONS RATIONALLY DEPENDENT ON THE EIGENPARAMETER. I

2002 ◽  
Vol 45 (3) ◽  
pp. 631-645 ◽  
Author(s):  
Paul A. Binding ◽  
Patrick J. Browne ◽  
Bruce A. Watson
Keyword(s):  
Boundary Conditions ◽  
Liouville Equation ◽  
Subject Classification ◽  
Asymptotics Of Eigenvalues ◽  
Primary 34B24 ◽  
Sturm Liouville

AbstractWe consider the Sturm–Liouville equation$$ -y''+qy=\lambda y\quad\text{on }[0,1], $$subject to the boundary conditions$$ y(0)\cos\alpha=y'(0)\sin\alpha,\quad\alpha\in[0,\pi), $$and$$\frac{y'}{y}(1)=a\lambda+b-\sum_{k=1}^N\frac{b_k}{\lambda-c_k}. $$Topics treated include existence and asymptotics of eigenvalues, oscillation of eigenfunctions, and transformations between such problems.AMS 2000 Mathematics subject classification: Primary 34B24; 34L20

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.

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