scholarly journals A Prüfer angle approach to singular Sturm–Liouville problems with Molčanov potentials

2007 ◽  
Vol 208 (1) ◽  
pp. 226-234 ◽  
Author(s):  
Paul Binding ◽  
Lyonell Boulton ◽  
Patrick J. Browne
Keyword(s):  
Prüfer Angle ◽  
Sturm Liouville

Sturm-Liouville Problems Whose Leading Coefficient Function Changes Sign

2003 ◽  
Vol 55 (4) ◽  
pp. 724-749 ◽  
Author(s):  
Xifang Cao ◽  
Qingkai Kong ◽  
Hongyou Wu ◽  
Anton Zettl
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Recent Result ◽  
Liouville Equation ◽  
Coefficient Function ◽  
Indexing Scheme ◽  
Number Of Zeros ◽  
Prüfer Angle ◽  
Sturm Liouville

AbstractFor a given Sturm-Liouville equation whose leading coefficient function changes sign, we establish inequalities among the eigenvalues for any coupled self-adjoint boundary condition and those for two corresponding separated self-adjoint boundary conditions. By a recent result of Binding and Volkmer, the eigenvalues (unbounded from both below and above) for a separated self-adjoint boundary condition can be numbered in terms of the Prüfer angle; and our inequalities can then be used to index the eigenvalues for any coupled self-adjoint boundary condition. Under this indexing scheme, we determine the discontinuities of each eigenvalue as a function on the space of such Sturm-Liouville problems, and its range as a function on the space of self-adjoint boundary conditions. We also relate this indexing scheme to the number of zeros of eigenfunctions. In addition, we characterize the discontinuities of each eigenvalue under a different indexing scheme.

scholarly journals Oscillation Properties of Singular Quantum Trees

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1266
Author(s):  
Monika Homa ◽  
Rostyslav Hryniv
Keyword(s):  
Differential Equations ◽  
Comparison Theorem ◽  
Interface Conditions ◽  
Oscillation Theorem ◽  
Prüfer Angle ◽  
Singular Coefficients ◽  
Sturm Liouville ◽  
Oscillation Theorems ◽  
Oscillation Properties ◽  
Sturm Theory

We discuss the possibility of generalizing the Sturm comparison and oscillation theorems to the case of singular quantum trees, that is, to Sturm-Liouville differential expressions with singular coefficients acting on metric trees and subject to some boundary and interface conditions. As there may exist non-trivial solutions of differential equations on metric trees that vanish identically on some edges, the classical Sturm theory cannot hold globally for quantum trees. However, we show that the comparison theorem holds under minimal assumptions and that the oscillation theorem holds generically, that is, for operators with simple spectra. We also introduce a special Prüfer angle, establish some properties of solutions in the non-generic case, and then extend the oscillation results to simple eigenvalues.

Application of two parameter eigencurves to Sturm–Liouville problems with eigenparameter-dependent boundary conditions

1995 ◽  
Vol 125 (6) ◽  
pp. 1205-1218 ◽  
Author(s):  
P. A. Binding ◽  
Patrick J. Browne
Keyword(s):  
Boundary Conditions ◽  
Asymptotic Theory ◽  
Liouville Problem ◽  
Prüfer Angle ◽  
Sturm Liouville ◽  
Sign Conditions ◽  
Image Position ◽  
Eigenvalue Dependent Boundary Conditions ◽  
Parameter Problem ◽  
Two Parameter

Oscillation, comparison and asymptotic theory for the Sturm-Liouville problemwith 1/p, q, r ε L1 ([0, 1]), p, r > 0, are studied subject to eigenvalue-dependent boundary conditionsThis continues previous work on cases with (− 1)j δj ≦ 0 where δj = ajdj − bjcj. We now consider the remaining sign conditions for δj, exploiting the interplay between the graph of cot θ− (λ, 1), for a modified Prüfer angle θ−, and the eigencurves of a related two-parameter problem.

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.

Sign in / Sign up

Export Citation Format

Share Document