arXiv subjects

Tom Cuchta

Publications and source records attributed to Tom Cuchta.

3 recordsLinked to original sources

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

Pr\"ufer 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 Pr\"ufer 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

Discrete Generating Series and Linear Difference Equations

We define discrete generating series for arbitrary functions \( f \colon \mathbb{Z}^n \rightarrow \mathbb{C} \) and derive functional relations that these series satisfy. For linear difference equations with constant coefficients, we establish explicit functional equations linking the generating series to the initial data, and for equations with polynomial coefficients, we introduce an analogue of Stanley's \( D \)-finiteness criterion, proving that a discrete generating series is \( D \)-finite if and only if the corresponding sequence is polynomially recursive. The framework is further generalized to multidimensional settings, where we investigate the interplay between discrete generating series and solutions to Cauchy problems for difference equations. Key structural properties are uncovered through the introduction of polynomial shift operators and projection techniques. The theory is illustrated with concrete examples, including the Tribonacci recurrence and Schr\"oder's second problem.

math.CA↗