arXiv subjects

F. Ayça Çetinkaya

Publications and source records attributed to F. Ayça Çetinkaya.

6 recordsLinked to original sources

arXiv2026

Transmutation operators for the Sturm--Liouville equation in impedance form with an integrable potential

In this work, we study a family of transmutation operators for the Sturm--Liouville operator in impedance form with an integrable potential satisfying a smallness condition. Specifically, we consider transmutations associated with the pair \[\textbf{L}_r=\frac{1}{r^2 (x)}\frac{d}{dx}r^2 (x)\frac{d}{dx} \quad \text{and} \quad \textbf{M}=\frac{d^2}{dx^2}. \] We extend existing results by showing that a particular transmutation operator on $W^{3,1} (-a,a)$ can be represented in integral form, where the kernel function satisfies a corresponding Goursat problem. We establish the existence of this kernel using two distinct approaches: the classical method of successive approximations and an application of Picard's theorem.

math.SP↗

arXiv2026

A Discrete Prüfer Transformation Approach to Sturm--Liouville Difference Equations and Eigenvalue Estimation

In this paper, we study regular second-order Sturm--Liouville difference equations using the discrete Prüfer transformation. By representing solutions in amplitude and phase coordinates, we analyze an exact algebraic phase system that guarantees unique, monotonic phase tracking and preserves classical oscillation properties. Using this theoretical foundation, we develop a Prüfer-based numerical shooting method to compute eigenvalues for discrete boundary value problems. To initialize the root-finding algorithm, we apply Gershgorin's theorem to the difference operator to establish mathematically guaranteed starting search intervals. Numerical experiments on classical benchmark problems demonstrate that the proposed method effectively isolates the discrete spectrum and converges to the exact continuous eigenvalues with second-order $\mathcal{O}(h^2)$ accuracy.

math.SP↗

arXiv2026

Inverse Quantum Potential Reconstruction via Generalized Bertlmann-Martin Inequalities

Reconstructing a radial (1D) quantum potential, V(r), from a few bound-state energies is a long-standing inverse problem because limited spectral data must constrain an entire potential. We present a Laplace-moment reconstruction pipeline that links the Bertlmann-Martin gap bound to generalized Bertlmann-Martin (GBM) even-moment ladders, continues the Laplace transform with Pade approximants, and inverts the transform to recover rho(r) and V(r). Odd moments are supplied by a physically consistent interpolation scheme. Benchmark settings and diagnostics for Coulomb, harmonic oscillator, Hulthen, Kratzer, and hyperbolic-well cases are stated so each approximation stage can be assessed under a common empirical basis. The conclusions are therefore limited to the reported benchmark settings rather than offered as universal method claims.

math.SP↗

arXiv2026

Nonlinear elliptic Dirichlet boundary value problems on time scales

We establish existence and uniqueness results for nonlinear elliptic Dirichlet boundary value problems on n-dimensional time scale domains. Time scales provide a unified framework that encompasses continuous, discrete, and hybrid settings. Under a Lipschitz condition on the nonlinearity bounded by the first eigenvalue, we prove existence and uniqueness using the contraction mapping theorem. Under a weaker one-sided growth condition, we establish existence using the Leray--Schauder fixed point theorem. To apply these functional analytic methods, we reformulate the problem as an operator equation, which requires developing the spectral theory for the Dirichlet Laplacian with mixed nabla-delta derivatives. We establish self-adjointness, positivity, and completeness of eigenfunctions, and the product eigenfunctions form a complete orthonormal basis in the n-dimensional setting.

math.AP↗

arXiv2025

Prüfer Transformation and Spectral Analysis for a Sturm--Liouville-Type Equation

We study a second-order differential equation involving a quasi-derivative, leading to a non-self-adjoint Sturm--Liouville-type problem with four coefficient functions. To analyze this equation, we develop a generalized Prüfer transformation that expresses solutions in terms of amplitude and phase variables. We further prove the monotonicity of eigenfunction zeros with respect to the spectral parameter and derive upper and lower bounds for the eigenvalues.

math.CA↗

arXiv2025

A Fractal Dirac Eigenvalue Problem: Spectral Properties and Numerical Examples

In this paper, we study a Dirac boundary value problem where the operator is considered with a derivative of order $α\in (0, 1]$, known as the $F^α$-derivative. We prove some spectral properties of eigenvalues and eigenfunctions and present numerical examples to demonstrate the practical implications of our approach.

math.SP↗