arXiv subjects

Łukasz Rzepnicki

Publications and source records attributed to Łukasz Rzepnicki.

3 recordsLinked to original sources

arXiv2025

Asymptotic behavior of solutions to the Dirac system with respect to a spectral parameter

We consider the Dirac system of ordinary differential equations \[ Y'(x) + \begin{bmatrix} 0 & σ_1(x) \\ σ_2(x) & 0 \end{bmatrix} Y(x) = iμ\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} Y(x), \quad Y(x) = \begin{bmatrix} y_1(x) \\ y_2(x) \end{bmatrix}, \] where $x \in [0,1]$, $μ\in \mathbb{C}$ is a spectral parameter, and $σ_j \in L^p[0,1],$ $j = 1,2,$ for $p \in [1,2).$ We study the asymptotic behavior of the system's fundamental solutions as $|μ| \to \infty$ in the half-plane $\operatorname{Im} μ> -r,$ where $r \geq 0$ is fixed, and obtain detailed asymptotic formulas. As an application, we derive new results on the half-plane asymptotics of fundamental solutions to Sturm--Liouville equations with singular potentials.

math.FA↗

arXiv2020

Asymptotic behavior of solutions of the Dirac system with an integrable potential

We consider the Dirac system on the interval $[0,1]$ with a spectral parameter $μ\in\mathbb{C}$ and a complex-valued potential with entries from $L_p[0,1]$, where $1\leq p <2$. We study the asymptotic behavior of its solutions in a stripe $|{\rm Im}\,μ|\le d$ for $μ\to \infty$. These results allows us to obtain sharp asymptotic formulas for eigenvalues and eigenfunctions of Sturm--Liouville operators associated with the aforementioned Dirac system.

math.SP↗

arXiv2020

On asymptotic behaviour of solutions of the Dirac system and applications to the Sturm-Liouville problem with a singular potential

The main focus of this paper is the following matrix Cauchy problem for the Dirac system on the interval $[0,1]:$ \[ D'(x)+\left[\begin{array}{cc} 0 & σ_1(x)\\ σ_2(x) & 0 \end{array} \right] D(x)=iμ\left[\begin{array}{cc} 1 & 0\\ 0 &-1 \end{array} \right]D(x),\quad D(0)=\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right], \] where $μ\in\mathbb{C}$ is a spectral parameter, and $σ_j\in L_2[0,1]$, $j=1,2$. We propose a new approach for the study of asymptotic behaviour of its solutions as $μ\to \infty$ and $|{\rm Im}\,μ|\le d$. As an application, we obtain new, sharp asymptotic formulas for eigenfunctions of Sturm-Liouville operators with singular potentials.

math.SP↗