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

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 ${\mathcal{L}}_{P,U}$: $$S_{m,P,U}(\mathbf{f})=\sum _{|n|\le m}[⟨\mathbf{f},{\mathbf{z}}_{2n}⟩{\mathbf{y}}_{2n}+⟨\mathbf{f},{\mathbf{z}}_{2n+1}⟩{\mathbf{y}}_{2n+1}].$$ 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 $\mathbf{f}(x)=\left(\begin{array}{c}{f}_1(x)\\ f_2(x)\end{array}\right)$, $f_1,f_2\in L_{\mu }[0,\pi ]$, $\mu \in [1,\infty ]$, the equiconvergence $$\Vert S_{m,P,U}(\mathbf{f})-S_{m,0,U}(\mathbf{f}){\Vert }_{L_{\nu }}\to 0\text{as}m\to \infty$$ in the norm of the space $(L_{\nu }[0,\pi ]{)}^2$, holds provided that $1/\varkappa +1/\mu -1/\nu \le 1$ (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 $S_{m,0,U}(\mathbf{f})$.

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