scholarly journals Asymptotics of Eigenvalues for Sturm-Liouville Problems with an Interior Singularity

1995 ◽  
Vol 116 (1) ◽  
pp. 88-118 ◽  
Author(s):  
B.J. Harris ◽  
D. Race
Keyword(s):  
Asymptotics Of Eigenvalues ◽  
Sturm Liouville ◽  
Interior Singularity

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 Reconstruction of potential function in inverse Sturm-Liouville problem via partial data

2021 ◽  
Vol 11 (2) ◽  
pp. 186-198
Author(s):  
Mehmet Açil ◽  
Ali Konuralp
Keyword(s):  
Boundary Conditions ◽  
Potential Function ◽  
Liouville Problem ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Asymptotics Of Eigenvalues ◽  
Time Saving ◽  
Partial Data ◽  
Sturm Liouville ◽  
Saving Algorithm

In this paper, three different uniqueness data are investigated to reconstruct the potential function in the Sturm-Liouville boundary value problem in the normal form. Taking account of R\"{o}hrl's objective function, the steepest descent method is used in the computation of potential functions. To decrease the volume of computation, we propose a theorem to precalculate the minimization parameter that is required in the optimization. Further, we propose a novel time-saving algorithm in which the obligation of using the asymptotics of eigenvalues and eigenfunctions and the appropriateness of selected boundary conditions are also eliminated. As partial data, we take two spectra, the set of the $j$th elements of the infinite numbers of spectra obtained by changing boundary conditions in the problem, and one spectrum with the set of terminal velocities. In order to show the efficiency of the proposed method, numerical results are given for three test potentials which are smooth, nonsmooth continuous, and noncontinuous, respectively.

scholarly journals Essentially isospectral transformations and their applications

2017 ◽  
Author(s):  
Namig J. Guliyev
Keyword(s):  
Boundary Conditions ◽  
Existence Theorems ◽  
Dirichlet Boundary Conditions ◽  
Trace Formulas ◽  
Asymptotics Of Eigenvalues ◽  
Regularized Trace ◽  
Eigenvalue Parameter ◽  
Nevanlinna Functions ◽  
Sturm Liouville ◽  
Uniqueness And Existence

We define and study the properties of Darboux-type transformations between Sturm–Liouville problems with boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary conditions). Using these transformations, we obtain various direct and inverse spectral results for these problems in a unified manner, such as asymptotics of eigenvalues and norming constants, oscillation of eigenfunctions, regularized trace formulas, and inverse uniqueness and existence theorems.

scholarly journals First regularized trace of the limit assignment of Sturm-Liouville type with two constant delays

Filomat ◽  
2015 ◽  
Vol 29 (1) ◽  
pp. 51-62 ◽  
Author(s):  
Natasa Pavlovic ◽  
Milenko Pikula ◽  
Biljana Vojvodic
Keyword(s):  
Characteristic Function ◽  
Liouville Type ◽  
Asymptotics Of Eigenvalues ◽  
Regularized Trace ◽  
Sturm Liouville ◽  
Spectral Assignment

We observe spectral assignment D2y = ?y defined by -y''(x) + q1(x)y(x - ?1) + q2(x)y(x - ?2) = ?y(x), ? = z2 (1) q1(x), q2(x) ? L1[0,?], ?1, ?2 ? (0,?) y(x -?1)? 0, x ? (0, ?1], ?1 = k0?2 (2), y(?) = 0 (3) In this paper, we construct a solution y(x, z) which satisfies (1) and (2), and then (3) is used to construct the characteristic function F(z), z ? C. Then the asymptotics of eigenvalues of the operator D2 is constructed. Finally, the first regularized trace is calculated.

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