The eigenvalue problem with interaction conditions at one interior singular point

Filomat ◽  
2017 ◽  
Vol 31 (17) ◽  
pp. 5411-5420 ◽  
Author(s):  
Oktay Mukhtarov ◽  
Kadriye Aydemir
Keyword(s):  
Singular Point ◽  
Eigenvalue Problem ◽  
Quantum Physics ◽  
Eigenvalue Problems ◽  
Physical Processes ◽  
Liouville Type ◽  
Vibrating String ◽  
Sturm Liouville ◽  
Transfer Problems ◽  
String Problems

Some physical processes, both classical physics and quantum physics reduced to eigenvalue problems for Sturm-Liouville equations. In the recent years there has been an increasing interest in discontinuous eigenvalue problems for various Sturm-Liouville type equations. Such problems are connected with heat transfer problems, vibrating string problems, diffraction problems and etc. In this study we shall investigate a class of two order eigenvalue problem with supplementary transmission conditions at one interior singular point. We give an operator-theoretic interpretation in suitable Hilbert space.

scholarly journals Asymptotic spectrum of multiparameter eigenvalue problems

1996 ◽  
Vol 39 (1) ◽  
pp. 119-132 ◽  
Author(s):  
Hans Volkmer
Keyword(s):  
Hilbert Space ◽  
Maximum Principle ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Asymptotic Spectrum ◽  
Multiparameter Eigenvalue Problem ◽  
Sturm Liouville

Results are given for the asymptotic spectrum of a multiparameter eigenvalue problem in Hilbert space. They are based on estimates for eigenvalues derived from the minim un-maximum principle. As an application, a multiparameter Sturm-Liouville problem is considered.

A high-speed method for eigenvalue problems. IV. Sturm-Liouville-type differential equations

1996 ◽  
Vol 96 (2-3) ◽  
pp. 247-262 ◽  
Author(s):  
T. Yano ◽  
K. Kitani ◽  
H. Miyatake ◽  
M. Otsuka ◽  
S. Tomiyoshi ◽  
...  
Keyword(s):  
Differential Equations ◽  
High Speed ◽  
Eigenvalue Problems ◽  
Liouville Type ◽  
Sturm Liouville ◽  
Speed Method

Non-standard oscillation theory for multiparameter eigenvalue problems

2012 ◽  
Vol 142 (4) ◽  
pp. 673-690
Author(s):  
P. A. Binding ◽  
H. Volkmer
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Oscillation Theory ◽  
Second Order ◽  
Order Derivative ◽  
Standard Case ◽  
Sturm Liouville ◽  
Two Parameters

An eigenvalue problem for k Sturm–Liouville equations coupled by k parameters λ1,…,λk is considered. In contrast to the standard case, for each r, the second-order derivative in the rth equation is multiplied by λr. This problem presents various interesting features. For example, the existence of eigenvalues with oscillation counts beyond a certain (computable) value is obtained without any of the restrictive definiteness conditions known from the standard case. Uniqueness is also analysed, and the results are given greater precision via eigencurve methods in the case of two equations coupled by two parameters.

scholarly journals Finding normal modes of a loaded string with the method of Lagrange multipliers

2021 ◽  
Author(s):  
Dong-Won Jung ◽  
Wooyong Han ◽  
U-Rae Kim ◽  
Jungil Lee ◽  
Chaehyun Yu ◽  
...  
Keyword(s):  
Quantum Physics ◽  
Eigenvalue Problems ◽  
Classical Mechanics ◽  
Normal Modes ◽  
Constraint Equation ◽  
Inverse Matrix ◽  
Eigenvalue Equation ◽  
Original Matrix ◽  
Vibrating String ◽  
The Difference

AbstractWe consider the normal mode problem of a vibrating string loaded with n identical beads of equal spacing, which involves an eigenvalue problem. Unlike the conventional approach to solving this problem by considering the difference equation for the components of the eigenvector, we modify the eigenvalue equation by introducing matrix-valued Lagrange undetermined multipliers, which regularize the secular equation and make the eigenvalue equation non-singular. Then, the eigenvector can be obtained from the regularized eigenvalue equation by multiplying the indeterminate eigenvalue equation by the inverse matrix. We find that the inverse matrix is nothing but the adjugate matrix of the original matrix in the secular determinant up to the determinant of the regularized matrix in the limit that the constraint equation vanishes. The components of the adjugate matrix can be represented in simple factorized forms. Finally, one can directly read off the eigenvector from the adjugate matrix. We expect this new method to be applicable to other eigenvalue problems involving more general forms of the tridiagonal matrices that appear in classical mechanics or quantum physics.

scholarly journals The asymptotic distribution of the eigenvalues of right definite multiparameter Sturm-Liouville systems

1993 ◽  
Vol 36 (1) ◽  
pp. 35-47 ◽  
Author(s):  
Bryan P. Rynne
Keyword(s):  
Eigenvalue Problem ◽  
Asymptotic Distribution ◽  
Type Equation ◽  
Liouville Type ◽  
Number Of Solutions ◽  
Multi Index ◽  
Asymptotic Eigenvalue ◽  
General Parameter ◽  
Sturm Liouville ◽  
Asymptotic Eigenvalue Distribution

This paper studies the asymptotic distribution of the multiparameter eigenvalues of a right definite multiparameter Sturm–Liouville eigenvalue problem. A uniform asymptotic analysis of the oscillation number of solutions of a single Sturm–Liouville type equation with potential depending on a general parameter is given; these results are then applied to the system of multiparameter Sturm–Liouville equations to give the asymptotic eigenvalue distribution for the system as a function of a “multi-index” oscillation number.

Homogeneous robin boundary conditions and discrete spectrum of fractional eigenvalue problem

2019 ◽  
Vol 22 (1) ◽  
pp. 78-94 ◽  
Author(s):  
Malgorzata Klimek
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Continuous Functions ◽  
Real Spectrum ◽  
Robin Boundary Conditions ◽  
Schmidt Operator ◽  
Sturm Liouville ◽  
Robin Boundary

Abstract We discuss a fractional eigenvalue problem with the fractional Sturm-Liouville operator mixing the left and right derivatives of order in the range (1/2, 1], subject to a variant of Robin boundary conditions. The considered differential fractional Sturm-Liouville problem (FSLP) is equivalent to an integral eigenvalue problem on the respective subspace of continuous functions. By applying the properties of the explicitly calculated integral Hilbert-Schmidt operator, we prove the existence of a purely atomic real spectrum for both eigenvalue problems. The orthogonal eigenfunctions’ systems coincide and constitute a basis in the corresponding weighted Hilbert space. An analogous result is obtained for the reflected fractional Sturm-Liouville problem.

scholarly journals Inverse Eigenvalue Problems for Singular Rank One Perturbations of a Sturm-Liouville Operator

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Xuewen Wu
Keyword(s):  
Eigenvalue Problem ◽  
Potential Function ◽  
Liouville Operator ◽  
Eigenvalue Problems ◽  
Inverse Eigenvalue Problem ◽  
Inverse Eigenvalue Problems ◽  
Rank One ◽  
Inverse Eigenvalue ◽  
Sturm Liouville

This paper is concerned with the inverse eigenvalue problem for singular rank one perturbations of a Sturm-Liouville operator. We determine uniquely the potential function from the spectra of the Sturm-Liouville operator and its rank one perturbations.

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