$L_\mu\to L_\nu$ equiconvergence of spectral decompositions for Dirac system with $L_\varkappa$ potential

TL;DR

This paper studies the convergence of spectral decompositions for a class of 1D Dirac operators with specific potential functions, establishing conditions under which the spectral series converge uniformly or in norm.

Contribution

It proves equiconvergence of spectral decompositions for Dirac operators with $p_1=p_4=0$ and $p_2,p_3$ in $L_ u$, extending classical results to less regular potentials.

Findings

Spectral decompositions converge in $L_ u$ norm under specified conditions.

Uniform pointwise convergence is achieved for certain $L_2$ potentials.

The results generalize classical spectral convergence to operators with less regular potentials.

Abstract

We consider 1d-Dirac operator $\mathcal L_{P,U}$ acting in $\mathbb H=(L_2[0,\pi])^2$ \begin{gather*} \ell(\mathbf y) = B\mathbf y + P(x)\mathbf y,\qquad B = \begin{pmatrix}-i&0\\0&i\end{pmatrix},\\ P(x) = \begin{pmatrix}p_1(x)&p_2(x)\\ p_3(x)&p_4(x) \end{pmatrix},\qquad\mathbf y = \begin{pmatrix}y_1(x)\\ y_2(x)\end{pmatrix} \end{gather*} with arbitrary regular boundary conditions $U(\mathbf y)=0$. The functions $p_j(x)$, $1\le j\le 4$, assumed to be complex valued and summable. Any regular 1d-Dirac operator of such kind has purely discrete spectrum $\{\lambda_n\}_{n\in\mathbb Z}$, $\lambda_n=n+O(1)$ as $|n|\to\infty$. We consider spectral decomposition $S_{P,U}(\mathbf f)=\lim_{m\to\infty}S_{m,P,U}(\mathbf f)$ associated with operator $\mathcal L_{P,U}$: $$ S_{m,P,U}(\mathbf f) = \sum_{|n|\le m}\Big[\langle\mathbf f,\mathbf z_{2n}\rangle\mathbf y_{2n} + \langle\mathbf f,\mathbf z_{2n+1}\rangle\mathbf y_{2n+1}\Big]. $$ Here $\mathbf y_n$ --- eigen- and associated functions of $\mathcal L_{P,U}$ and $\{\mathbf z_n\}_{n\in\mathbb Z}$ --- biorthogonal in $\mathbb H$ system. Our main result claims that for every regular 1d-Dirac operator with $p_1=p_4=0$ and $p_2,\,p_3\in L_\varkappa[0,\pi]$, $\varkappa\in(1,\infty]$, and for every function $\mathbf f(x)=\begin{pmatrix}f_1(x)\\ f_2(x)\end{pmatrix}$, $f_1,\,f_2\in L_\mu[0,\pi]$, $\mu\in[1,\infty]$, the equiconvergence $$ \|S_{m,P,U}(\mathbf f)-S_{m,0,U}(\mathbf f)\|_{L_\nu}\to0\quad \text{as}\ m\to\infty $$ in the norm of the space $(L_\nu[0,\pi])^2$, holds provided that $1/\varkappa+1/\mu-1/\nu\le1$ (with one exception $\varkappa=\nu=\infty$, $\mu=1$). In particular, in the case $\varkappa=\mu=2$, $\nu=\infty$ we obtain uniform pointwise equiconvergence on $[0,\pi]$ of the series $S_{m,P,U}(\mathbf f)$ and $S_{m,0,U}(\mathbf f)$.