Zeros Of Eigenfunctions Research Articles

Ambarzumyan theorem by zeros of eigenfunction

Abstract In this short paper, we give the proof of the Ambarzumyan theorem by zeros of eigenfunctions (nodal points) different from eigenvalues for the one-dimensional p-Laplacian eigenvalue problem. We show that the potential function q(x) is zero if the spectrum is in the specific form. We consider this theorem for p-Laplacian equation with periodic and anti-periodic cases. If p = 0, results are coincided with the results given for Sturm-Liouvile problem.

RETRACTED ARTICLE: On eigenstructure of q-Bernstein operators

The quantum analogue of Bernstein operators $${\mathcal {B}}_{m,q}$$ reproduce the linear polynomials which are therefore eigenfunctions corresponding to the eigenvalue $$1,~\forall ~ q>0$$ . In this article the rest of eigenstructure of q-Bernstein operators and the distinct behaviour of zeros of eigenfunctions for cases (i) $$1>q>0$$ , and (ii) $$q>1$$ are discussed. Graphical analysis for some eigenfunctions and their roots are presented with the help of MATLAB. Also, matrix representation for diagonalisation of q-Bernstein operators is discussed.

On the integral characteristic function of the Sturm-Liouville problem

We introduce a function whose zeros, and only these zeros, are eigenvalues of the corresponding Sturm-Liouville problem. The boundary conditions of the problem depend continuously on the spectral parameter. Therefore, it makes sense to call the function thus constructed a characteristic function of the Sturm-Liouville problem (however, it is not a characteristic function in the ordinary sense). An investigation of the function thus obtained enables us to prove the solvability of the problem in question, to find the asymptotic behaviour of the eigenvalues, to obtain comparison theorems, and to introduce an indexing of the eigenvalues and the zeros of eigenfunctions in a natural way. Bibliography: 31 titles.

Inverse nodal problem for polynomial pencil of Sturm‐Liouville operator

The paper is about boundary value problem for polynomial pencil of Sturm‐Liouville operators. Especially, we find all coefficients of the operator by using nodal points (zeros of eigenfunctions). Regularly, we find eigenvalues, nodal points, and nodal lengths by Prüfer substitution. These results are used to give a reconstruction formula for all complex functions qd(x), , which are known potentials in the theory. However, method is similar with some papers; our results more general then because of including many potential functions.

Reconstruction of the Volterra-type integro-differential operator from nodal points

In this work, the Sturm–Liouville problem perturbated by a Volterra-type integro-differential operator is studied. We give a uniqueness theorem and an algorithm to reconstruct the potential of the problem from nodal points (zeros of eigenfunctions).

Solving inverse Sturm–Liouville problem with separated boundary conditions by using two different input data

ABSTRACTIn this study, we consider Sturm–Liouville (SL) equation under the separated boundary conditions on a finite interval. We get the approximate solutions of inverse SL problem by using different input data as eigenvalues and nodes (zeros of eigenfunctions), separately, and calculate the computed errors related to the obtained approximate solutions. To calculate the approximate solutions, we use Chebyshev interpolation technique by applying first kind of Chebyshev polynomials. Eventually, the numerical results are presented by providing some examples.

Some Ambarzumyan Type Theorems for Bessel Operator on a Finite Interval

In this study, we deal with an inverse problem for Bessel operator on a finite interval. We present some results of the associated with Ambarzumyan’s theorem by using spectrum and nodal points (zeros of eigenfunction).

Recovering the Sturm-Liouville Operator with Singular Potential Using Nodal Data

The paper deals with the Sturm-Liouville operator with singular potential. We assume that the potential is a sum of an a priori known distribution from a certain class and an unknown sufficiently smooth function. The inverse problem is to recover the operator using zeros of eigenfunctions (nodes) as an input data. For this inverse problem we obtain a procedure for constructing the solution.

The inverse nodal problem for a differential operator with an eigenvalue in the boundary condition

We consider the Sturm–Liouville problem with an eigenvalue dependent boundary condition. In this work, by using method of Yang [X.F. Yang, A solution of the inverse nodal problem, Inverse Problems 13 (1997) 203–213.], we reconstruct the potential of the Sturm–Liouville problem with an eigenvalue in the boundary condition from nodal points (zeros of eigenfunctions). Also, we give a uniqueness theorem.

Nodal lines, ergodicity and complex numbers

This article reviews two rigorous results about the complex zeros of eigenfunctions of the Laplacian, that is, the zeros of the analytic continuation of the eigenfunctions to the complexification of the underlying space. Such a complexification of the problem is analogous to studying the complex zeros of polynomials with real coefficients. The first result determines the limit distribution of complex zeros of `ergodic eigenfunctions' such as eigenfunctions of classically chaotic systems. The second result determines the expected distribution of complex zeros for complexifications of Gaussian random waves adapted to the Riemannian manifold. The resulting distribution is the same in both cases. It is singular along the set of real points.

Sign Regularity Conditions for Discontinuous Boundary-Value Problems

For a multipoint discontinuous boundary-value problem with nonoscillating differential operator, we present an analog of the Kalafati conditions that ensure the sign regularity property, i.e., the property that the number of sign changes of the solution does not exceed the number of sign changes of the function on the right-hand side. The sign regularity property allows one to verify whether the spectrum of the corresponding spectral problem exhibits the Sturm properties (i.e., the reality, positiveness, and simplicity of eigenvalues, the alternation of zeros of eigenfunctions, etc.).

Sturm-Liouville Problems Whose Leading Coefficient Function Changes Sign

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.