scholarly journals On the missing eigenvalue problem for an inverse Sturm–Liouville problem

2009 ◽  
Vol 91 (5) ◽  
pp. 468-475 ◽  
Author(s):  
Guangsheng Wei ◽  
Hong-Kun Xu
Keyword(s):  
Eigenvalue Problem ◽  
Liouville Problem ◽  
Sturm Liouville

X.—The Two Parameter Sturm-Liouville Problem for Ordinary Differential Equations

1971 ◽  
Vol 69 (2) ◽  
pp. 139-148
Author(s):  
B. D. Sleeman
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Liouville Problem ◽  
Sturm Liouville ◽  
Image Position ◽  
Two Parameter ◽  
General Conditions

SynopsisThis paper discusses the existence, under fairly general conditions, of solutions of the two-parameter eigenvalue problem denned by the differential equation,and three point Sturm-Liouville boundary conditions.

Expansions in terms of sets of functions with complex eigenvalues

1948 ◽  
Vol 44 (2) ◽  
pp. 242-250 ◽  
Author(s):  
R. Peierls
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Real Function ◽  
Liouville Problem ◽  
Nuclear Collisions ◽  
Complex Eigenvalues ◽  
Sturm Liouville ◽  
Image Position ◽  
Unusual Type

In the following I discuss the properties, in particular the completeness of the set of eigenfunctions, of an eigenvalue problem which differs from the well-known Sturm-Liouville problem by the boundary condition being of a rather unusual type.The problem arises in the theory of nuclear collisions, and for our present purpose we take it in the simplified formwhere 0 ≤ x ≤ 1. V(x) is a given real function, which we assume to be integrable and to remain between the bounds ± M, and W is an eigenvalue. The eigenfunction ψ(x) is subject to the boundary conditionsand

Large eigenvalues of a Sturm-Liouville problem

1967 ◽  
Vol 63 (2) ◽  
pp. 473-475 ◽  
Author(s):  
J. H. E. Cohn
Keyword(s):  
Eigenvalue Problem ◽  
Bounded Function ◽  
Liouville Problem ◽  
Sturm Liouville ◽  
Image Position

Consider the eigenvalue problemwhere q(x) is a (bounded) function in the class L(a, b). We may suppose without loss of generality that 0 ≤ α < π and 0 ≤ β < π. Then as is well known there are infinitely many eigenvalues λr (r = 0, 1, 2, …) and λn ∽ n2π2 (b − a)−2 as n → ∞.

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.

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 Convolution algebras arising from Sturm-Liouville transforms and applications

2001 ◽  
Vol 27 (4) ◽  
pp. 221-228
Author(s):  
Jason P. Huffman ◽  
Henry E. Heatherly
Keyword(s):  
Eigenvalue Problem ◽  
Integral Transform ◽  
Operational Calculus ◽  
Liouville Problem ◽  
Generalized Convolution ◽  
Convolution Operation ◽  
Convolution Algebras ◽  
Sturm Liouville ◽  
Linear Integral ◽  
Differential And Integral Equations

A regular Sturm-Liouville eigenvalue problem gives rise to a related linear integral transform. Churchill has shown how such an integral transform yields, under certain circumstances, a generalized convolution operation. In this paper, we study the properties of convolution algebras arising in this fashion from a regular Sturm-Liouville problem. We give applications of these convolution algebras for solving certain differential and integral equations, and we outline an operational calculus for classes of such equations.

A fractional analysis in higher dimensions for the Sturm-Liouville problem

2021 ◽  
Vol 24 (2) ◽  
pp. 585-620
Author(s):  
Milton Ferreira ◽  
M. Manuela Rodrigues ◽  
Nelson Vieira
Keyword(s):  
Eigenvalue Problem ◽  
Clifford Analysis ◽  
Fractional Derivatives ◽  
Infinite Sequence ◽  
Variational Calculus ◽  
Liouville Problem ◽  
Higher Dimensions ◽  
Fractional Analysis ◽  
Sturm Liouville ◽  
Left And Right

Abstract In this work, we consider the n-dimensional fractional Sturm-Liouville eigenvalue problem, by using fractional versions of the gradient operator involving left and right Riemann-Liouville fractional derivatives. We study the main properties of the eigenfunctions and the eigenvalues of the associated fractional boundary problem. More precisely, we show that the eigenfunctions are orthogonal and the eigenvalues are real and simple. Moreover, using techniques from fractional variational calculus, we prove in the main result that the eigenvalues are separated and form an infinite sequence, where the eigenvalues can be ordered according to increasing magnitude. Finally, a connection with Clifford analysis is established.

scholarly journals A trace formula for discontinuous eigenvalue problem

Filomat ◽  
2017 ◽  
Vol 31 (8) ◽  
pp. 2425-2431
Author(s):  
Fatma Hıra ◽  
Nihat Altınışık
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Trace Formula ◽  
Liouville Problem ◽  
Regularized Trace ◽  
Sturm Liouville

In this paper, we deal with a Sturm-Liouville problem which has discontinuity at one point and contains an eigenparameter in a boundary condition. We obtain a regularized trace formula for the problem.

scholarly journals On the equivalence of discrete Sturm–Liouville problem with nonlocal boundary conditions to the algebraic eigenvalue problem

2015 ◽  
Vol 56 ◽  
pp. 66-71
Author(s):  
Jurij Novickij ◽  
Artūras Štikonas
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Solvability Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Sturm Liouville ◽  
Algebraic Eigenvalue Problem

We consider the finite difference approximation of the second order Sturm–Liouville equation with nonlocal boundary conditions (NBC). We investigate the condition when the discrete Sturm–Liouville problem can be transformed to an algebraic eigenvalue problem and denote this condition as solvability condition. The examples of the solvability for the most popular NBCs are provided. The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/ 2014).

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.

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