scholarly journals The Properties of Eigenvalues and Eigenfunctions for Nonlocal Sturm–Liouville Problems

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 820
Author(s):  
Zhiwen Liu ◽  
Jiangang Qi
Keyword(s):  
Spectral Properties ◽  
Eigenvalue Problems ◽  
Finite Interval ◽  
Essential Difference ◽  
Nonlocal Problems ◽  
Comparison Result ◽  
Eigenvalues And Eigenfunctions ◽  
Comparison Results ◽  
Sturm Liouville ◽  
Oscillation Properties

The present paper is concerned with the spectral theory of nonlocal Sturm–Liouville eigenvalue problems on a finite interval. The continuity, differentiability and comparison results of eigenvalues with respect to the nonlocal potentials are studied, and the oscillation properties of eigenfunctions are investigated. The comparison result of eigenvalues and the oscillation properties of eigenfunctions indicate that the spectral properties of nonlocal problems are very different from those of classical Sturm–Liouville problems. Some examples are given to explain this essential difference.

scholarly journals Fundamental Spectral Theory of Fractional Singular Sturm-Liouville Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Erdal Bas
Keyword(s):  
Spectral Theory ◽  
Liouville Operator ◽  
Spectral Properties ◽  
Liouville Problem ◽  
Eigenvalues And Eigenfunctions ◽  
Sturm Liouville

We give the theory of spectral properties for eigenvalues and eigenfunctions of Bessel type of fractional singular Sturm-Liouville problem. We show that the eigenvalues and eigenfunctions of the problem are real and orthogonal, respectively. Furthermore, we prove new approximations about the topic.

scholarly journals A New Approach for Linear Eigenvalue Problems and Nonlinear Euler Buckling Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Meltem Evrenosoglu Adiyaman ◽  
Sennur Somali
Keyword(s):  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Numerical Results ◽  
High Order Accuracy ◽  
Order Accuracy ◽  
Eigenvalues And Eigenfunctions ◽  
New Approach ◽  
Euler Buckling ◽  
Large Step ◽  
Sturm Liouville

We propose a numerical Taylor's Decomposition method to compute approximate eigenvalues and eigenfunctions for regular Sturm-Liouville eigenvalue problem and nonlinear Euler buckling problem very accurately for relatively large step sizes. For regular Sturm-Liouville problem, the technique is illustrated with three examples and the numerical results show that the approximate eigenvalues are obtained with high-order accuracy without using any correction, and they are compared with the results of other methods. The numerical results of Euler Buckling problem are compared with theoretical aspects, and it is seen that they agree with each other.

Sturm–Liouville problems with indefinite weights and Everitt's inequality

1996 ◽  
Vol 126 (5) ◽  
pp. 1097-1112 ◽  
Author(s):  
Hans Volkmer
Keyword(s):  
Spectral Properties ◽  
Eigenvalue Problems ◽  
Baire Category ◽  
Integral Inequalities ◽  
Sufficient Condition ◽  
Indefinite Weights ◽  
Sturm Liouville ◽  
Indefinite Problems

It is shown that spectral properties of Sturm–Liouville eigenvalue problems with indefinite weights are related to integral inequalities studied by Everitt. A result of Beals on indefinite problems leads to a sufficient condition for the validity of such an inequality. A Baire category argument is used to show that, in general, the inequality under consideration does not hold.

The anharmonic oscillator

1978 ◽  
Vol 360 (1703) ◽  
pp. 575-586 ◽  
Keyword(s):  
Eigenvalue Problems ◽  
Anharmonic Oscillator ◽  
Eigenvalues And Eigenfunctions ◽  
Transition Moments ◽  
Comparable Accuracy ◽  
Oscillation Properties

Accurate eigenvalues and eigenfunctions of the anharm onic oscillator ( H = p 2 + x 2 + λx 4 , λ > 0) and the quartic oscillator ( H = p 2 + x 4 ) are obtained in all regimes of the quantum num ber n and the anharm onicity constant λ. Transition moments of comparable accuracy are obtained for the quartic oscillator. The method, applicable quite generally for eigenvalue problems, is non-perturbative and involves the use of an appropriately scaled basis for the determ ination of each eigenvalue. The appropriate scaling formula for a given regime of ( n , λ) is constructed from the oscillation properties of the eigenfunctions. More general anharm onic oscillators are also discussed.

scholarly journals Transmission problem for the Sturm-Liouville equation involving a retarded argument

Filomat ◽  
2021 ◽  
Vol 35 (6) ◽  
pp. 2071-2080
Author(s):  
Erdoğan Şen
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Spectral Parameter ◽  
Liouville Equation ◽  
Spectral Properties ◽  
Asymptotic Formulas ◽  
Retarded Argument ◽  
Eigenvalues And Eigenfunctions ◽  
Discontinuous Boundary ◽  
Sturm Liouville

In this work, spectral properties of a discontinuous boundary-value problem with retarded argument which contains a spectral parameter in the boundary conditions and in the transmission conditions at the point of discontinuity are investigated. To this aim, asymptotic formulas for the eigenvalues and eigenfunctions are obtained.

Spectral properties of a beam equation with eigenvalue parameter occurring linearly in the boundary conditions

2021 ◽  
pp. 1-22
Author(s):  
Ziyatkhan S. Aliyev ◽  
Gunay T. Mamedova
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Spectral Properties ◽  
Sufficient Conditions ◽  
Beam Equation ◽  
Asymptotic Formulas ◽  
Eigenvalues And Eigenfunctions ◽  
Root Functions ◽  
Eigenvalue Parameter ◽  
Oscillation Properties

In this paper, we consider an eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary conditions. The location of eigenvalues on real axis, the structure of root subspaces and the oscillation properties of eigenfunctions of this problem are investigated, and asymptotic formulas for the eigenvalues and eigenfunctions are found. Next, by the use of these properties, we establish sufficient conditions for subsystems of root functions of the considered problem to form a basis in the space $L_p,1 < p < \infty$ .

Asymptotics of Sturm-Liouville eigenvalues for problems on a finite interval with one limit-circle singularity, I

1984 ◽  
Vol 99 (1-2) ◽  
pp. 51-70 ◽  
Author(s):  
F. V. Atkinson ◽  
C. T. Fulton
Keyword(s):  
Asymptotic Expansion ◽  
Eigenvalue Problem ◽  
Hypergeometric Function ◽  
Finite Interval ◽  
Confluent Hypergeometric Function ◽  
Limit Circle ◽  
Asymptotic Formulae ◽  
Sturm Liouville

SynopsisAsymptotic formulae for the positive eigenvalues of a limit-circle eigenvalue problem for –y” + qy = λy on the finite interval (0, b] are obtained for potentials q which are limit circle and non-oscillatory at x = 0, under the assumption xq(x)∈L1(0,6). Potentials of the form q(x) = C/xk, 0<fc<2, are included. In the case where k = 1, an independent check based on the limit-circle theory of Fulton and an asymptotic expansion of the confluent hypergeometric function, M(a, b; z), verifies the main result.

scholarly journals Symbolic Algorithm of the Functional-Discrete Method for a Sturm–Liouville Problem with a Polynomial Potential

2018 ◽  
Vol 18 (4) ◽  
pp. 703-715 ◽  
Author(s):  
Volodymyr Makarov ◽  
Nataliia Romaniuk
Keyword(s):  
Finite Interval ◽  
General Scheme ◽  
Liouville Problem ◽  
Piecewise Constant Function ◽  
Polynomial Potential ◽  
Discrete Method ◽  
Piecewise Constant ◽  
Sturm Liouville ◽  
Symbolic Algorithm ◽  
Theoretical Results

AbstractA new symbolic algorithmic implementation of the general scheme of the exponentially convergent functional-discrete method is developed and justified for the Sturm–Liouville problem on a finite interval for the Schrödinger equation with a polynomial potential and the boundary conditions of Dirichlet type. The algorithm of the general scheme of our method is developed when the potential function is approximated by the piecewise-constant function. Our algorithm is symbolic and operates with the decomposition coefficients of the eigenfunction corrections in some basis. The number of summands in these decompositions depends on the degree of the potential polynomial and on the correction number. Our method uses the algebraic operations only and does not need solutions of any boundary value problems and computations of any integrals unlike the previous version. A numerical example illustrates the theoretical results.

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