A new approach to uniqueness for inverse Sturm--Liouville problems on finite intervals

Abstract

In this paper, an approach for studying inverse Sturm-Liouville problems with integrable potentials on finite intervals is presented. We find the relations between Weyl solutions and mj -functions of Sturm-Liouville problems, and by finding the connection between these and the solutions of second-order partial differential equations for transformation kernels associated with Sturm-Liouville operators, we prove the uniqueness of the solution of inverse problems.

On the solution of an inverse Sturm–Liouville problem with a delay andeigenparameter-dependent boundary conditions

“…Then, for sufficiently large values of n , C ′ (η) ̸ = 0 and η is in the proximity of n 2 . Hence, the proof is completed by Rolle's theorem and (11). 2 Relation (10) has a central role to approximate the eigenvalues of L .…”

“…Boundary value problems consisting of (1) with α = 1 without eigenparameters in boundary conditions have been studied since the 1930s (for example, see [2,[9][10][11]17,18], and for more details see also [4,7]). In [8,15], inverse Sturm-Liouville problems with boundary conditions depending on an eigenparameter without delay parameter were investigated.…”

Determination of a differential pencil from interior spectral data on a union of two closed intervals

“…They proved that the spectrum of the problem −y + q(x)y = λy, t ∈ (0, 1) y (0) − hy(0) = y (1) + Hy(1) = 0 and the logarithmic derivatives of the eigenfunctions at the point 1/2 uniquely determine the potential q(x) on the whole interval [0, 1] almost everywhere. This kind of problems for the differential operators on a continuous interval were studied in [8]- [14], [18,22,23], [28]- [34].…”

A New Type of Sturm-Liouville Equation in the Non-Newtonian Calculus

“…The one-dimensional timeindependent Schrödinger equation in quantum mechanics can be given as an example of SL equation. Significant results have been obtained by many mathematicians over the years regarding the SL equation (see [25][26][27][28][29][30][31][32][33][34][35][36]). This equation has not yet been addressed in multiplicative calculus.…”